{"id":"3","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle C A B = 90 \\circ$$, D and E are points on sides $$A C$$ and $$A B$$ respectively, and $$A D = A E$$. Connect $$B D$$ and $$C E$$. Let $$A F \\bot C E$$ intersect at point G and intersect $$B C$$ at point F. Let $$F P \\bot B D$$ intersect $$C E$$ at point H and intersect $$A B$$ at point P. Connect $$A H$$. If $$A H$$ bisects $$\\angle B A F$$, then the following conclusions: ① $$A P = B E$$; ② $$\\angle H P E = \\angle H E P$$; ③ $$A P + A D = 2 A G$$; ④ $$C H = A F + F H$$; ⑤ $$\\angle D B C = 15 \\circ$$. The correct conclusions are numbered"},{"type":"image_path","image_path":"images/3_q0.png"}],"answer":"①②③④⑤"}
{"id":"6","difficulty":"0.2","question_list":[{"type":"text","text":"Definition: In a triangle, if the measure of one interior angle is three times the measure of another interior angle, then such a triangle is called a \"beautiful triangle.\" For example: a triangle with interior angles of $$100 \\circ$$, $$60 \\circ$$, and $$20 \\circ$$ is a \"beautiful triangle.\" As shown in the figure, point $$D$$ lies on side $$AB$$ of $$\\triangle ABC$$, connect $$DC$$, with $$\\angle BDC > 90^\\circ$$. Construct the angle bisector of $$\\angle ADC$$, intersecting $$AC$$ at point $$E$$. On $$DC$$, take a point $$F$$ such that $$\\angle EFC + \\angle BDC = 180^\\circ$$ and $$\\angle DEF = \\angle B$$. If $$\\triangle BCD$$ is a \"beautiful triangle\", then $$\\angle B$$ equals        ."},{"type":"image_path","image_path":"images/6_q0.png"}],"answer":"$$36 \\circ$$"}
{"id":"11","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$ and $$\\triangle D E F$$, $$\\angle A C B = \\angle D F E = 90 \\circ$$, $$\\angle A = \\angle E D F = 60 \\circ$$, and $$E D$$ moves along the line $$A B$$. If $$A C = E D = 2$$, then the minimum value of $$C E + C F$$ is      ."},{"type":"image_path","image_path":"images/11_q0.png"}],"answer":"$$\\sqrt{21}$$"}
{"id":"18","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 4$$, point $$F$$ lies on $$B C$$ with $$B F = 1$$, and point $$E$$ moves along segment $$A B$$ (not coinciding with $$A$$ or $$B$$). Segment $$E F$$ is rotated clockwise around point $$F$$ by $$30^{\\circ}$$ to obtain $$F G$$, and $$C G$$ is connected."},{"type":"image_path","image_path":"images/18_q0.png"},{"type":"text","text":"(1) When $$B E = \\sqrt{3}$$, then $$C G =$$            ; (2) During the motion of $$E$$, the minimum value of $$C G$$ is          ."}],"answer":"$$\\sqrt{13}$$     $$1 + \\frac{3 \\sqrt{3}}{2}$$"}
{"id":"20","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 2$$, $$B D \\bot A C$$, point E lies on line $$B D$$, $$B E = A C$$, $$\\angle C E A = 90 \\circ$$, then the minimum value of $$B C$$ is        ."},{"type":"image_path","image_path":"images/20_q0.png"}],"answer":"$$\\sqrt{5} - 1$$"}
{"id":"30","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = 2$$, $$B C = 3$$, $$\\angle B = 60^{\\circ}$$, and $$P$$ is a moving point on side $$B C$$ ($$B P > 1$$). Fold $$\\triangle A B P$$ along $$A P$$ to obtain $$\\triangle A B^{'} P$$. The ray $$P B^{'}$$ intersects the ray $$A D$$ at point $$E$$. The following statements are given: (1) When $$A B^{'} \\bot A B$$, $$B^{'} A = B^{'} E$$; (2) When point $$B^{'}$$ lies on $$A D$$, quadrilateral $$A B P B^{'}$$ is a rhombus; (3) As point $$P$$ moves, the minimum value of segment $$A E$$ is 2; (4) Connect $$B B^{'}$$, then the area of quadrilateral $$A B P B^{'}$$ is always equal to $$\\frac{1}{2} A P \\cdot B B^{'}$$. The correct statement(s) are     ."},{"type":"image_path","image_path":"images/30_q0.png"}],"answer":"(1)(2)(4)"}
{"id":"31","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle B = 90 \\circ$$, $$\\angle B C D = 45 \\circ$$. Connect $$A C$$ and from point D draw a perpendicular to diagonal $$A C$$, intersecting $$B C$$ at point E and $$A C$$ at point F. If $$A B = B E$$, $$\\angle D A C = 2 \\angle A C B$$, $$C D = 3 \\sqrt{2}$$, then the length of $$A C$$ is      ."},{"type":"image_path","image_path":"images/31_q0.png"}],"answer":"$$2 \\sqrt{10}$$"}
{"id":"32","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 3$$, point $$M$$ is the midpoint of $$A B$$, and point $$N$$ is a moving point on side $$C D$$ (not coinciding with either endpoint). If quadrilateral $$A M N D$$ is folded along line $$M N$$ to obtain quadrilateral $$E F M N$$ (point $$E$$ and point $$F$$ correspond to point $$D$$ and point $$A$$, respectively), connect $$E C$$ and $$F C$$. When $$E C \\bot F C$$, the perimeter of $$\\triangle E C F$$ is      ."},{"type":"image_path","image_path":"images/32_q0.png"}],"answer":"$$\\frac{15 + 9 \\sqrt{5}}{5}$$"}
{"id":"35","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point $$H$$ is the midpoint of $$C D$$, and lines $$B D$$ and $$A H$$ intersect at point $$G$$. From point $$A$$, draw a perpendicular to $$B D$$, intersecting $$B D$$ at point $$E$$, and extend $$A E$$ to intersect $$B C$$ at point $$F$$. If $$A B = B G$$, take a point $$P$$ on $$B D$$ (not coinciding with $$B$$ or $$D$$) such that line $$C P$$ passes through the midpoint of one side of quadrilateral $$D E F H$$. Find the value of $$B P : B D$$       ."},{"type":"image_path","image_path":"images/35_q0.png"}],"answer":"$$\\frac{13}{18}$$ or $$\\frac{5}{7}$$ or $$\\frac{10}{27}$$"}
{"id":"42","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B = 90 \\circ$$, $$A B = 4$$, $$B C = 3$$, first fold $$\\triangle A B C$$ along $$A C$$ to $$\\triangle A B^{'} C$$, then fold $$\\triangle A B^{'} C$$ along $$A B^{'}$$ to $$\\triangle A B^{'} C^{'}$$, draw $$C D \\parallel A B$$ through point $$C$$ intersecting $$A C^{'}$$ at point $$D$$, then the length of $$C D$$ is          ."},{"type":"image_path","image_path":"images/42_q0.png"}],"answer":"$$\\frac{200}{39}$$"}
{"id":"46","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square ABCD$$, $$AB = 2$$, $$BC = 4$$, $$\\angle B = 30^\\circ$$, point $$P$$ is a moving point on side $$BC$$, connect $$AP$$, rotate segment $$AP$$ clockwise about point $$P$$ by $$90^\\circ$$; when the image point $$E$$ of point $$A$$ lies exactly on the line containing one of the sides of $$\\square ABCD$$, the length of segment $$AP$$ is      ."},{"type":"image_path","image_path":"images/46_q0.png"}],"answer":"$$1$$ or $$2$$ or $$\\sqrt{2}$$."}
{"id":"47","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the square $$ABCD$$ has side length 6, and points $$E$$ and $$F$$ are on sides $$BC$$ and $$CD$$ respectively, with $$BE = CF = 2$$. Connect $$AE$$ and $$AF$$. The perpendicular bisector of $$AE$$ intersects $$AB$$, $$AE$$, $$AF$$, and $$CD$$ at points $$G$$, $$M$$, $$N$$, and $$H$$, respectively. Find the length of $$MN$$      ."},{"type":"image_path","image_path":"images/47_q0.png"}],"answer":"$$\\frac{7}{9} \\sqrt{10}$$"}
{"id":"49","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the radius of $$\\bigodot O$$ is $$2 \\sqrt{3}$$, $$AB$$ is the diameter of $$\\bigodot O$$, a line $$DE \\bot OB$$ is drawn through the midpoint $$C$$ of radius $$OB$$, intersecting $$\\bigodot O$$ at points $$D$$ and $$E$$, point $$F$$ lies on $$\\bigodot O$$, connect $$FD$$, draw $$HA \\bot AB$$ from point $$A$$ intersecting the extension of $$FD$$ at $$H$$, if $$DH = 3 \\sqrt{7}$$, then the length of segment $$AH$$ is        , and the length of segment $$EF$$ is           ."},{"type":"image_path","image_path":"images/49_q0.png"}],"answer":"9     $$\\frac{12 \\sqrt{7}}{7}$$"}
{"id":"62","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$AB$$ is the diameter of $$\\bigodot O$$, $$BC$$ and $$CD$$ are tangent lines to $$\\bigodot O$$, with points of tangency at $$B$$ and $$D$$, respectively. Point $$E$$ is a moving point on $$AB$$. Connect $$CE$$ and $$DE$$. If $$AB = 2\\sqrt{5}$$ and $$BC = 2$$, then the minimum value of $$CE + DE$$ is          ."},{"type":"image_path","image_path":"images/62_q0.png"}],"answer":"$$\\frac{14}{3}$$"}
{"id":"78","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the diameter of semicircle O is $$MN$$, point A lies on radius $$OM$$, B is the midpoint of arc $$\\overset{⌢}{MN}$$, and point C lies on arc $$BN$$. A rectangle $$ABCD$$ is constructed with $$AB$$ and $$BC$$ as adjacent sides. Side $$CD$$ intersects $$MN$$ at point E. Connect $$DO$$ and extend $$DO$$ to intersect $$AB$$ at point P. If $$BP = 2AP$$, then the value of $$\\frac{BC}{AB}$$ is                      ."},{"type":"image_path","image_path":"images/78_q0.png"}],"answer":"$$\\frac{\\sqrt{5}}{3}$$/$$\\frac{1}{3} \\sqrt{5}$$"}
{"id":"85","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A C B$$, $$\\angle A C B = 90 \\circ, A C = B C = 6, C D = 2, C H \\bot B D$$ at H, point O is the midpoint of $$A B$$, connect $$O H$$, then $$O H =$$      ."},{"type":"image_path","image_path":"images/85_q0.png"}],"answer":"$$\\frac{6}{5} \\sqrt{5}$$"}
{"id":"86","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle with side length 2. Fold $$\\triangle A B C$$ along the line $$A C$$ to obtain $$\\triangle A B^{'} C$$, then translate $$\\triangle A B^{'} C$$ along the line $$A C$$ to obtain $$\\triangle A^{'} B^{' '} C^{'}$$. The minimum perimeter of $$\\triangle B B^{' '} C^{'}$$ is       ."},{"type":"image_path","image_path":"images/86_q0.png"}],"answer":"$$2 \\sqrt{7} + 2$$"}
{"id":"87","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4. Take the midpoint $$E$$ of side $$A B$$, connect $$C E$$, draw $$B F \\bot C E$$ through point $$B$$, intersecting at point $$F$$, connect $$D F$$, draw $$A H \\bot D F$$ through point $$A$$, intersecting at point $$H$$, intersecting $$C E$$ at point $$M$$, and intersecting $$B C$$ at point $$N$$. Then (1) $$E F =$$                 ; (2) $$M N =$$                 ."},{"type":"image_path","image_path":"images/87_q0.png"}],"answer":"$$\\frac{2 \\sqrt{5}}{5}$$     1"}
{"id":"94","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure: In quadrilateral $$A B C D$$, $$\\angle B A C = 90 \\circ$$, $$\\angle A D C =\\text{45} \\circ$$, point $$C$$ moves along segment $$D G$$, and $$\\angle A B D = 2 \\angle B D C$$, $$B D$$ intersects $$A C$$ at point $$E$$. If $$B D = 9$$, $$C B = 3 \\sqrt{5}$$, $$A G$$ intersects $$D B$$ at point $$F$$, in triangle $$A D G$$, $$t a n \\angle D A F = 1$$, then the sine of $$\\angle B E A$$ is        ."},{"type":"image_path","image_path":"images/94_q0.png"}],"answer":"$$\\frac{3}{5}$$ or $$\\frac{12}{13}$$"}
{"id":"108","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the rhombus $$A B C D$$ has side length $$6$$, $$\\angle A B C = 60^\\circ$$, and segment $$A C$$ is connected. Points $$E$$ and $$F$$ are moving points on segments $$A B$$ and $$A C$$ respectively (not coinciding with endpoints), and $$B E = A F$$. $$B F$$ intersects $$C E$$ at point $$P$$. Extend $$B F$$ to intersect side $$A D$$ (or side $$C D$$) at point $$G$$. The following conclusions are given: ① $$\\triangle A B F \\sim \\triangle B C E$$; ② When $$B E = 4$$, $$B E : C G = 4 : 3$$; ③ Connect $$E F$$, the minimum area of quadrilateral $$B C F E$$ is $$6 \\sqrt{3}$$; ④ When $$\\angle P D C$$ is maximized, the length of segment $$D P$$ is $$6 \\sqrt{2}$$. All correct conclusions are        ."},{"type":"image_path","image_path":"images/108_q0.png"}],"answer":"①②④"}
{"id":"111","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 60 \\circ$$, $$A B = 5$$, $$A C = 8$$, P is a point inside $$\\triangle A B C$$, $$\\angle B P C = 120 \\circ$$, connect $$A P$$, then the minimum length of $$A P$$ is           ."},{"type":"image_path","image_path":"images/111_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"115","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, given equilateral triangle $$\\triangle A B C$$, $$A C = 4$$, construct square $$A C D E$$ with side $$A C$$ (points $$A$$, $$C$$, $$D$$, $$E$$ arranged counterclockwise), and let the extensions of $$B C$$ and $$E D$$ intersect at point $$F$$. Point $$P$$ starts from point $$B$$ and moves along $$B F$$ toward point $$F$$, stopping at $$F$$. Point $$Q$$ moves along segment $$A C$$ and $$C D$$, always satisfying that $$P Q$$ is perpendicular to a side of the square. Connect $$E Q$$, $$E P$$, and $$P Q$$. When $$\\frac{E Q}{P Q} = \\frac{5}{\\sqrt{3}}$$, the area of $$\\triangle E P Q$$ is                     ."},{"type":"image_path","image_path":"images/115_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{2}$$ or $$\\frac{17 \\sqrt{123} - 99 \\sqrt{3}}{16}$$ or $$\\frac{4 \\sqrt{141} - 12}{11}$$"}
{"id":"117","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$\\angle B A D = 120^\\circ$$, $$A B = 2$$, $$B C = 3$$, point E is a moving point on side $$B C$$, connect $$A E$$, rotate $$A E$$ clockwise around point E by $$60^\\circ$$ to obtain $$F E$$, connect $$C F$$ and $$D F$$, then the minimum value of $$C F + D F$$ is       ."},{"type":"image_path","image_path":"images/117_q0.png"}],"answer":"$$\\sqrt{7}$$"}
{"id":"118","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$D$$ is the point dividing $$A B$$ into four equal parts closer to point $$B$$, connect $$C D$$, and take the midpoint $$E$$ of $$C D$$, connect $$B E$$. If $$\\angle A C D = 30 \\circ$$, $$\\angle B E D = 60 \\circ$$, $$C D = 2$$, then the length of $$A B$$ is      ."},{"type":"image_path","image_path":"images/118_q0.png"}],"answer":"$$\\frac{2 \\sqrt{31}}{3}$$"}
{"id":"124","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B D$$ and $$\\triangle C B D$$ are both equilateral triangles. Draw a ray inside $$\\angle A B C$$, and construct point $$E$$, the reflection of point $$C$$ over line $$B M$$. Connect $$A E$$ and extend it to intersect $$B M$$ at point $$F$$. Connect $$C E$$ and $$C F$$. If $$A E = 5$$ and $$E F = 2$$, then the length of $$B F$$ is      ."},{"type":"image_path","image_path":"images/124_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"126","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an inscribed triangle of $$\\bigodot O$$, $$A C > B C$$, $$\\angle A C B = 45 \\circ$$, and $$\\triangle A B C$$ is rotated counterclockwise about point A to obtain $$\\triangle A D E$$ (with points B and C corresponding to points D and E, respectively). When $$A D$$ is tangent to $$\\bigodot O$$, the line containing $$D E$$ is also tangent to $$\\bigodot O$$. If the radius of $$\\bigodot O$$ is 3, then the length of $$B C$$ is      ."},{"type":"image_path","image_path":"images/126_q0.png"}],"answer":"$$4 - \\sqrt{2}$$/$$- \\sqrt{2} + 4$$"}
{"id":"129","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B F$$, C and E are points on sides $$B F$$ and $$A F$$ respectively. Connect $$A C$$ and $$B E$$, intersecting at point D. If $$A B = A C = A E = 12$$, $$A D = 9$$, $$\\angle B A C = 60^\\circ$$, then the length of segment $$E F$$ is          ."},{"type":"image_path","image_path":"images/129_q0.png"}],"answer":"7.5/$$\\frac{15}{2}$$"}
{"id":"130","difficulty":"0.2","question_list":[{"type":"text","text":"Given in $$\\triangle ABO$$, $$\\angle B = 90^\\circ$$, $$\\angle AOB = 30^\\circ$$, rotate $$\\triangle ABO$$ around point $$O$$ by $$120^\\circ$$ and $$240^\\circ$$ to obtain $$\\triangle DCO$$ and $$\\triangle FEO$$, respectively. Connect $$AD$$, $$AF$$, $$FD$$, intersecting $$OC$$, $$OB$$, $$OE$$ at points $$P$$, $$Q$$, $$R$$ respectively. If the area of $$\\triangle ADF$$ is $$10$$, then the area of $$\\triangle PQR$$ is     ."},{"type":"image_path","image_path":"images/130_q0.png"}],"answer":"$$\\frac{10}{3}$$/$$3 \\frac{1}{3}$$"}
{"id":"132","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\triangle ABC$$, $$AB = BC$$, point $$D$$ is a moving point on side $$AC$$, and an isosceles right triangle $$\\triangle DBE$$ is constructed with $$BD$$ as one leg. $$DE$$ intersects $$BC$$ at point $$F$$, and $$CE$$ is connected. From point $$B$$, draw $$BQ \\bot DE$$, intersecting at point $$P$$ and intersecting $$CD$$ at point $$Q$$. Among the following conclusions, the correct ones are ______.\n\n① $$\\triangle ABD \\sim \\triangle CBE$$;\n② $$AD^{2} + CQ^{2} = DQ^{2}$$;\n③ When $$AD : DC = 1 : 2$$, $$S_{\\triangle BEC} + S_{\\triangle DCE} = S_{\\triangle DBE}$$;\n④ When $$CD = BC$$, $$BD : EF = \\sqrt{2} + 1$$."},{"type":"image_path","image_path":"images/132_q0.png"}],"answer":"①②④"}
{"id":"136","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$D E$$ bisects $$\\angle A D C$$, intersecting $$B C$$ at point E, $$E F \\bot A E$$, intersecting $$C D$$ at point F. Construct rectangle $$A E F G$$ with sides $$A E$$ and $$E F$$, and $$F G$$ intersects $$D A$$ at point H. If $$C E = 3$$, $$A H = 4$$, then $$A E =$$                ."},{"type":"image_path","image_path":"images/136_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"140","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is inscribed in $$\\bigodot O$$, $$A B$$ is the diameter of $$\\bigodot O$$, chord $$C D = C B$$, $$C E \\bot A B$$ at point $$E$$, connect $$B D$$ intersecting $$C E$$ at point $$F$$, intersecting $$A C$$ at $$G$$. If $$C D = 4 \\sqrt{5}$$, $$A C = 8 \\sqrt{5}$$, which of the following statements are correct?  \n① $$\\angle A C B = 90 \\circ$$; ② $$C F = B F$$; ③ The area of $$\\triangle A B G$$ is $$100$$; ④ $$B D = 16$$."},{"type":"image_path","image_path":"images/140_q0.png"}],"answer":"$$\\textcircled{1} \\textcircled{2} \\textcircled{4}$$"}
{"id":"141","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A C$$ is a diagonal, $$A E \\bot B C$$ at point E, point F is a point on the extension of $$A E$$, and $$\\angle A C F = \\angle C A F$$, the extensions of the line segments intersect at point G. If $$A B = \\sqrt{5}$$, $$A D = 4$$, $$\\text{tan} \\angle A B C = 2$$, then the length of $$B G$$ is          ."},{"type":"image_path","image_path":"images/141_q0.png"}],"answer":"$$\\frac{20 \\sqrt{5}}{19}$$/$$\\frac{20}{19} \\sqrt{5}$$"}
{"id":"142","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, the altitude $$A D$$ bisects side $$B C$$ exactly, $$\\angle B = 30^\\circ$$, point $$P$$ is a point on the extension of $$B A$$, and point $$O$$ is a point on segment $$A D$$ such that $$O P = O C$$. The following conclusions: $$\\textcircled{1} A B = A C$$, $$\\textcircled{2} \\triangle A O P \\cong \\triangle A O C$$, $$\\textcircled{3} \\angle A P O + \\angle D C O = 30^\\circ$$, $$\\textcircled{4} \\triangle O P C$$ is an equilateral triangle. Among these, the correct ones are ______. (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/142_q0.png"}],"answer":"$$\\textcircled{1}\\textcircled{3}\\textcircled{4}$$"}
{"id":"145","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, given that the side length of square $$A B C D$$ is 8, point $$E$$ is a moving point on side $$A D$$, connect $$E C$$, rotate $$E C$$ counterclockwise by $$90^{\\circ}$$ about point $$E$$ to obtain $$E F$$, connect $$D F$$ and $$C F$$, then the minimum value of $$D F + C F$$ is        ."},{"type":"image_path","image_path":"images/145_q0.png"}],"answer":"$$8 \\sqrt{5}$$"}
{"id":"151","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, and $$AB = 2\\sqrt{2}$$. Point $$C$$ lies on the semicircle, and $$OC \\bot AB$$, with the foot of the perpendicular at point $$O$$. Point $$P$$ is any point on the semicircle. From point $$P$$, draw $$PE \\bot OC$$, intersecting at point $$E$$. Let the incenter of $$\\triangle OPE$$ be $$M$$. Connect $$OM$$ and $$PM$$. As point $$P$$ moves along the semicircle from point $$B$$ to point $$A$$, the length of the path traced by the incenter $$M$$ is      ."},{"type":"image_path","image_path":"images/151_q0.png"}],"answer":"$$\\pi$$"}
{"id":"153","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 5$$, point $$E$$ lies on side $$A B$$ such that $$B E = 2$$, and point $$P$$ is a moving point inside the rectangle satisfying $$\\angle P A B = \\angle P B C$$. Connect $$P E$$, rotate $$P E$$ counterclockwise by $$90^\\circ$$ about point $$P$$ and extend it to point $$P F$$ such that $$P F = 2 P E$$. Connect $$C F$$. Find the minimum value of $$C F$$          ."},{"type":"image_path","image_path":"images/153_q0.png"}],"answer":"$$3 \\sqrt{5} - 3 \\sqrt{2}$$"}
{"id":"157","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$ with side length 3, point D is a moving point on side $$BC$$, satisfying $$CE \\bot AD$$, with foot at point E. Point F is the symmetric point of point B with respect to point E. Point G is the trisection point on $$AB$$ closer to point A. Connect $$FG$$. When $$\\angle AGF$$ is maximized, $$FG =$$            ."},{"type":"image_path","image_path":"images/157_q0.png"}],"answer":"2"}
{"id":"159","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$B D$$ and $$A C$$ intersect at point $$O$$. Point $$D$$ is rotated clockwise around point $$A$$ by $$60^\\circ$$ to obtain point $$D^{'}$$. Connect $$O D^{'}$$ and $$C D^{'}$$. When the length of segment $$O D^{'}$$ is minimized, the length of $$C D^{'}$$ is $$\\sqrt{3} - 1$$. Find the side length of the rhombus."},{"type":"image_path","image_path":"images/159_q0.png"}],"answer":"$$\\sqrt{2}$$"}
{"id":"170","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, $$E$$ and $$F$$ are points on $$C D$$ and $$B C$$ respectively, $$\\angle E A F = 45^\\circ$$, $$A E$$ and $$A F$$ intersect $$B D$$ at $$H$$ and $$G$$ respectively, $$A G \\bot A I$$, $$D I \\bot B D$$. Which of the following conclusions is correct? (Fill in the serial number).\n\n① $$B F + D E = E F$$, ② $$A H^{2} + A G^{2} = O G^{2} + O H^{2} + A D^{2}$$, ③ $$B G^{2} + D H^{2} = G H^{2}$$, ④ $$G H \\cdot B D = \\frac{1}{2} A H \\cdot G I$$, ⑤ $$B G^{2} + D G^{2} = B D^{2}$$"},{"type":"image_path","image_path":"images/170_q0.png"}],"answer":"①②③"}
{"id":"179","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle B C P$$, $$\\angle B P C = 90 \\circ$$, A lies on side $$P C$$, $$\\angle B A C = 120 \\circ$$, D is the midpoint of $$B C$$, $$A D = 4 \\sqrt{3}$$, then the maximum value of segment $$P D$$ is       ."},{"type":"image_path","image_path":"images/179_q0.png"}],"answer":"12"}
{"id":"187","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, point $$D$$ is a moving point on side $$A C$$, and point $$E$$ is a point on side $$B C$$ such that $$A D = C E$$. Connect $$A E$$ and $$B D$$. When the length of segment $$C F$$ is minimized, the value of $$\\frac{S_{\\triangle A B F}}{S_{\\triangle A B C}}$$ is      ."},{"type":"image_path","image_path":"images/187_q0.png"}],"answer":"$$\\frac{1}{3}$$"}
{"id":"191","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, A, F, B, C are four points on circle O, quadrilateral $$O A B C$$ is a parallelogram, $$\\angle F A B = 15^\\circ$$, connect $$O F$$ intersecting $$A B$$ at point E, draw the tangent to the circle at point C intersecting the extension of $$A B$$ at point D and the extension of $$A F$$ at point H. If $$O A = 3$$, then $$C H =$$     ."},{"type":"image_path","image_path":"images/191_q0.png"}],"answer":"$$9 - 3 \\sqrt{3}$$/$$- 3 \\sqrt{3} + 9$$"}
{"id":"193","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the altitude from vertex $$A$$ to side $$BC$$ in equilateral $$\\triangle ABC$$, and point $$E$$ is a moving point on $$AD$$ (point $$E$$ does not coincide with point $$A$$). Connect $$CE$$, and construct an equilateral $$\\triangle CEF$$ with $$CE$$ as a side. If $$AB = 6$$, then the minimum value of the length of segment $$DF + CF$$ is      ."},{"type":"image_path","image_path":"images/193_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"207","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, draw $$D E \\bot A D$$ intersecting diagonal $$A C$$ at point $$E$$, and connect $$B E$$. Point $$P$$ is a moving point on segment $$B E$$, and let $$P^{'}$$ be the reflection of point $$P$$ across line $$D E$$. Point $$Q$$ is a moving point on $$A C$$, and connect $$P^{'} Q$$ and $$D Q$$. If $$A E = 9$$ and $$C E = 7$$, then the maximum value of $$D Q - P^{'} Q$$ is      ."},{"type":"image_path","image_path":"images/207_q0.png"}],"answer":"$$\\frac{8 \\sqrt{2}}{3}$$/$$\\frac{8}{3} \\sqrt{2}$$"}
{"id":"212","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$ABCD$$ is 4, and points E and F lie on sides $$AD$$ and $$BC$$, respectively. Folding quadrilateral $$ABFE$$ along $$EF$$ yields quadrilateral $$EFNM$$, and the image of point A, denoted M, lies exactly on line $$CD$$. If $$DM = 1$$, then the length of segment $$BF$$ is    ."},{"type":"image_path","image_path":"images/212_q0.png"}],"answer":"$$\\frac{9}{8}$$ or $$\\frac{25}{8}$$"}
{"id":"213","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = \\sqrt{2}, A D = 2$$, $$E$$ is the midpoint of side $$A D$$, point $$F$$ lies on side $$C D$$, connect $$E F$$, fold $$\\triangle D E F$$ along $$E F$$, and let the image of point $$D$$ be $$D^{'}$$, connect $$B D^{'}$$. If $$B D^{'} = 2$$, then $$D F =$$      ."},{"type":"image_path","image_path":"images/213_q0.png"}],"answer":"$$\\sqrt{3} - \\sqrt{2}$$/$$- \\sqrt{2} + \\sqrt{3}$$"}
{"id":"216","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the three vertices $P$, $Q$, and $R$ of the equilateral triangle $PQR$ lie on three sides $AD$, $AB$, and $DC$ of the square $ABCD$, respectively. Given that the side length of square $ABCD$ is $4\\sqrt{6}$, the sum of the lengths of $AQ$ and $DR$ is             ."},{"type":"image_path","image_path":"images/216_q0.png"}],"answer":"$$12 \\sqrt{2}$$"}
{"id":"218","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, a semicircle $$O$$ is drawn with $$B C$$ as its diameter, and an arc $$\\overset{⌢}{A C}$$ is drawn with center $$D$$ and radius $$D A$$, intersecting the semicircle $$O$$ at point $$P$$. We call point $$P$$ a \"wonderful point\" of square $$A B C D$$. Many line segments passing through the wonderful point have beautiful properties, both in terms of position and quantity, worth exploring. Connect $$P A$$, $$P B$$, $$P C$$, $$P D$$, and extend $$P D$$ to intersect $$A B$$ at point $$F$$. Among the following conclusions: ① $$F D = F B + B C$$; ② $$\\angle A P C = 135 \\circ$$; ③ $$S_{\\triangle P B C} = \\frac{1}{2} A P^{2}$$; ④ $$\\text{tan} \\angle B A P = \\frac{1}{3}$$; the correct conclusion(s) is/are      ."},{"type":"image_path","image_path":"images/218_q0.png"}],"answer":"①②③④"}
{"id":"223","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 2$$, point $$P$$ is a point on side $$A B$$, and point $$A$$ is symmetric to point $$A^{'}$$ with respect to line $$P D$$. The following conclusions: ① When line $$A^{'} D$$ divides the area of the rectangle in the ratio $$1 : 3$$, $$A P = 2 \\sqrt{2} - 2$$; ② When points $$P$$, $$A^{'}$$, and $$C$$ are collinear, $$A P = 4 - 2 \\sqrt{3}$$; ③ Let the length of segment $$A^{'} B$$ be $$d$$, then $$2 \\sqrt{5} - 2 \\leq d \\leq 2 \\sqrt{5} + 2$$; ④ Let the area of $$\\triangle A^{'} A B$$ be $$S$$, then $$0 < S \\leq \\frac{32}{5}$$. Among these, the correct ones are      . (Only fill in the serial numbers)"},{"type":"image_path","image_path":"images/223_q0.png"}],"answer":"②④"}
{"id":"225","difficulty":"0.2","question_list":[{"type":"text","text":"In quadrilateral $$A B C D$$, $$\\angle A = \\angle D = 90 \\circ$$, $$C D = B C$$, the following four judgments:"},{"type":"image_path","image_path":"images/225_q0.png"},{"type":"text","text":"① If $$\\angle C = 120 \\circ$$, then $$A D = \\frac{1}{2} B C$$;\n② Connect $$A C$$, $$D B$$, if $$A C$$ bisects $$D B$$ perpendicularly, then $$A D = B C$$;\n③ Connect $$A C$$, construct $$\\angle D A C = \\angle A C B$$, then quadrilateral $$A B C D$$ is a square;\n④ The reflection of point $$A$$ over line $$B D$$ must lie on line $$B C$$.\nAmong these, the correct serial numbers are        . (Write all correct serial numbers)"}],"answer":"②③④"}
{"id":"230","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, construct squares $$A B E F$$, $$B C G H$$, and $$A C M N$$ outwardly on the three sides of the triangle. Draw $$B I \\bot E H$$ with foot at point I, and extend $$I B$$ to intersect $$A C$$ at point J. The following conclusions are given: ① $$A B = M G$$.② $$S_{\\triangle B E H} = S_{\\triangle A F N}$$.③ Draw $$B I \\bot E H$$ with foot at point I, and extend $$I B$$ to intersect $$A C$$ at point J, then $$A J = C J$$.④ If J is the midpoint of $$A C$$, then $$2 B J = E H$$.Which of the above conclusions are correct? (Only fill in the serial numbers)"},{"type":"image_path","image_path":"images/230_q0.png"}],"answer":"①②③④"}
{"id":"231","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is $$2 \\sqrt{5}$$, point $$E$$ is the midpoint of $$C D$$, $$B E$$ intersects $$A C$$ at point $$M$$, $$F$$ is a point on $$A D$$, connect $$B F$$ which intersects $$A C$$ and $$A E$$ at points $$G$$ and $$H$$ respectively, and $$B F \\bot A E$$, connect $$M H$$, then $$M H =$$        ."},{"type":"image_path","image_path":"images/231_q0.png"}],"answer":"$$\\frac{2 \\sqrt{13}}{3}$$"}
{"id":"235","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, $$B C = 9$$, point D is a point on side $$B C$$ such that $$B D = 6$$, draw $$D E \\bot A B$$ with foot at point E, connect $$A D$$, then $$A D =$$    ; point F is the midpoint of $$A D$$, connect $$C F$$, draw $$F G \\bot C F$$ intersecting $$D E$$ at point G, then $$F G =$$     ."},{"type":"image_path","image_path":"images/235_q0.png"}],"answer":"$$3 \\sqrt{7}$$     $$\\frac{\\sqrt{39}}{2}$$"}
{"id":"237","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the square $$ABCD$$ with side length 4, point $$E$$ is the midpoint of $$BC$$, and point $$F$$ is a moving point on $$AB$$. Fold $$\\triangle BEF$$ along $$EF$$ to obtain $$\\triangle GEF$$. Connect $$GC$$, and construct $$\\triangle GHC$$ as the reflection of $$\\triangle GEC$$ across $$GC$$. Connect $$DG$$ and $$DH$$. When $$\\triangle DGH$$ is an isosceles triangle, the length of $$BF$$ is        ."},{"type":"image_path","image_path":"images/237_q0.png"}],"answer":"2 or $$2 \\sqrt{3}$$"}
{"id":"265","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A D = 3$$, $$A C = 6$$, point $$E$$ is the midpoint of $$A B$$, point $$F$$ is a point on diagonal $$A C$$, $$\\triangle G E F$$ is symmetric to $$\\triangle A E F$$ with respect to line $$E F$$, $$E G$$ intersects $$A C$$ at point $$H$$, when one of the interior angles of $$\\triangle C G H$$ is $$90 \\circ$$, then the length of $$C G$$ is      ."},{"type":"image_path","image_path":"images/265_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{2}$$ or $$\\frac{3 \\sqrt{7}}{2}$$"}
{"id":"266","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, point E is a moving point on $$AB$$, point F is on $$CD$$, and $$AE = CF$$. From point B, draw $$BG \\bot EF$$ intersecting $$AD$$ at point G, with foot at point M:"},{"type":"image_path","image_path":"images/266_q0.png"},{"type":"text","text":"(1) When point G is the midpoint of $$AD$$, the length of $$AE$$ is            ; (2) Connect $$AM$$, then the minimum value of $$AM$$ is           ;"}],"answer":"1     $$\\sqrt{10} - \\sqrt{2}$$/$$- \\sqrt{2} + \\sqrt{10}$$"}
{"id":"279","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point $$E$$ is a point on side $$BC$$, connect $$DE$$, point $$F$$ is the midpoint of $$DE$$, draw a perpendicular to $$DE$$ through point $$F$$, intersecting $$AB$$ and $$CD$$ at points $$M$$ and $$N$$ respectively, connect $$AC$$ intersecting $$MN$$ at point $$G$$. If $$\\angle DNG = 60^\\circ$$ and $$AB = 3$$, then the length of $$FG$$ is $$      $$"},{"type":"image_path","image_path":"images/279_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"283","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$B C = 6$$, points $$E$$ and $$H$$ are the midpoints of sides $$A D$$ and $$B C$$ respectively. Connect $$B E$$, and let point $$F$$ be a moving point on $$B E$$. Connect $$H F$$ and $$D F$$; extend $$D F$$ to intersect $$A B$$ at point $$P$$. If $$P B = P F$$, then the length of $$H F$$ is        ."},{"type":"image_path","image_path":"images/283_q0.png"}],"answer":"3"}
{"id":"291","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, O is the intersection point of the diagonals of rectangle $$A B C D$$, point E lies on side $$A D$$, connect $$O E$$, rotate segment $$O E$$ counterclockwise around point O by $$90 \\circ$$ to obtain segment $$O F$$ (point F lies inside rectangle $$A B C D$$), connect $$A F$$ and $$E F$$. If $$A B = 2$$, $$A D = 4$$, then the maximum area of $$\\triangle A E F$$ is       ."},{"type":"image_path","image_path":"images/291_q0.png"}],"answer":"$$\\frac{9}{8}$$"}
{"id":"294","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$D$$ is the midpoint of $$A B$$, and connect $$C D$$. Fold $$\\triangle B C D$$ along $$C D$$ to obtain $$\\triangle E C D$$, and connect $$A E$$. If $$A C \\bot D E$$ at point $$F$$, and $$B C = 4$$, then the length of $$A F$$ is        ."},{"type":"image_path","image_path":"images/294_q0.png"}],"answer":"$$\\frac{4 \\sqrt{10}}{5}$$"}
{"id":"298","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the square $$ABCD$$ has side length 4, and points $$E$$ and $$F$$ lie on sides $$BC$$ and $$CD$$ respectively. $$AE$$ bisects $$\\angle BAC$$. Connect $$BF$$, intersecting $$AE$$ and $$AC$$ at points $$G$$ and $$H$$ respectively, and $$AE = BF$$. The following four conclusions are given: ① $$AE$$ bisects $$BH$$ perpendicularly; ② If point $$P$$ is a moving point on side $$AB$$, then the minimum value of $$PH + PC$$ is $$4\\sqrt{3}$$; ③ $$GH^{2} = AG \\cdot EG$$; ④ $$S_{\\triangle ABH} = 6\\sqrt{2}$$. Among these, the correct ones are             ."},{"type":"image_path","image_path":"images/298_q0.png"}],"answer":"①②③"}
{"id":"307","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, $$A B = 4$$, point D moves along side $$A C$$ from C to A, point E moves along side $$B C$$ from B to C, and $$C D = B E$$. Connect $$B D$$ and $$A E$$, intersecting at point P. Rotate side $$A C$$ clockwise 90° about point C to obtain $$C M$$, and on ray $$C M$$, take segment $$C F$$ such that $$C F = \\sqrt{3} A C$$. During the motion of D and E, find the minimum value of $$\\frac{1}{2} P C + P F$$      ."},{"type":"image_path","image_path":"images/307_q0.png"}],"answer":"$$2 \\sqrt{21}$$"}
{"id":"308","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$B C = 4$$, $$A B = 8$$, point D is a moving point on side $$A C$$, and a square $$B D E F$$ is constructed with $$B D$$ as a side above $$B D$$. Then the minimum value of $$A E$$ is                , and at this time the length of $$C D$$ is                   ."},{"type":"image_path","image_path":"images/308_q0.png"}],"answer":"$$6 - 2 \\sqrt{3}$$/$$- 2 \\sqrt{3} + 6$$     $$2 \\sqrt{3} - 2$$/$$- 2 + 2 \\sqrt{3}$$"}
{"id":"309","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$\\angle A B C = 120 \\circ$$, $$A B = 10$$. Connect $$B D$$, and $$B D \\bot C D$$. Let $$C E$$ bisect $$\\angle D C B$$ and intersect $$A D$$ at point $$E$$. Point $$N$$ lies on side $$B C$$, and $$B C = 4 C N$$. If segment $$P Q$$ (with point $$P$$ to the left of point $$Q$$) moves along segment $$C E$$, with $$P Q = 5 \\sqrt{3}$$, connect $$B P$$ and $$N Q$$. Find the minimum value of $$B P + P Q + Q N$$."},{"type":"image_path","image_path":"images/309_q0.png"}],"answer":"$$5 \\sqrt{7} + 5 \\sqrt{3}$$"}
{"id":"317","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, $$D$$ is a point on side $$B C$$, and connect $$A D$$. Construct $$\\text{Rt} \\triangle A E C$$ with hypotenuse $$A C$$ to the right of side $$A C$$, and connect $$B E$$. Let $$F$$ be a point on $$B E$$ such that $$B F = \\frac{2}{3} B E$$, and connect $$D F$$. If $$A B = 6$$ and $$C D = 1$$, when $$D F$$ takes its minimum value, the area of $$\\triangle B D F$$ is           ."},{"type":"image_path","image_path":"images/317_q0.png"}],"answer":"$$\\frac{5 \\sqrt{3}}{2} - \\frac{5 \\sqrt{21}}{7}$$"}
{"id":"318","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 60 \\circ$$, point $$D$$ lies on $$A B$$, $$C D = 14$$, $$\\angle B D C = 60 \\circ$$, extend $$C B$$ to point $$E$$ such that $$C E = A C$$, draw $$E F \\bot C D$$ through point $$E$$, intersecting $$C D$$ at point $$F$$ and intersecting $$A B$$ at point $$G$$, if $$2 D G = A D$$, then $$D F =$$        ."},{"type":"image_path","image_path":"images/318_q0.png"}],"answer":"$$\\frac{7}{5}$$"}
{"id":"330","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C \\sim \\triangle B D C$$, $$\\angle A = \\angle C B D = 90 \\circ$$, $$A B = 4$$, $$B E$$ bisects $$\\angle C B D$$ and intersects $$C D$$ at point E, and intersects the extension of $$A C$$ at point F. Connect $$D F$$. If the triangle with vertices B, C, E is similar to $$\\triangle D E F$$, then $$A C =$$      ."},{"type":"image_path","image_path":"images/330_q0.png"}],"answer":"2 or $$4 \\sqrt{2} - 4$$"}
{"id":"334","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, two congruent right triangles $$BCE$$ and $$DAF$$ are constructed outwardly with $$BC$$ and $$AD$$ as the hypotenuses of square $$ABCD$$. From point C, draw $$CG \\bot AF$$ intersecting at point G and intersecting $$AD$$ at point H. From point B, draw $$BI \\bot CG$$ intersecting at point I. From point D, draw $$DK \\bot BE$$ intersecting the extension of $$EB$$ at point K and intersecting $$CG$$ at point L. If $$S_{\\text{quadrilateral } ABI G} = 2 S_{\\triangle BCE}$$ and $$GH = 1$$, then the length of $$DK$$ is    ."},{"type":"image_path","image_path":"images/334_q0.png"}],"answer":"$$\\frac{15}{2}$$"}
{"id":"335","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle B = \\angle D = 90 \\circ$$, $$A B = 2$$, $$A D = 3$$, points $$M$$, $$N$$ lie on sides $$B C$$, $$C D$$ respectively. When $$\\angle A M N + \\angle A N M = 120 \\circ$$, the perimeter of $$\\triangle A M N$$ is minimized. Find its minimum perimeter."},{"type":"image_path","image_path":"images/335_q0.png"}],"answer":"$$2 \\sqrt{19}$$"}
{"id":"340","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, it is given that quadrilateral $$ABCD$$ is a rectangle, $$AB = 8$$, $$AD = 6$$, $$F$$ is a moving point on side $$BC$$, $$O$$ is the midpoint of $$AC$$, $$OE \\bot OF$$ intersects $$AB$$ at $$E$$, and connect $$EF$$, $$OB$$. If $$OB$$ divides the area of $$\\triangle OEF$$ into two parts in the ratio $$1:2$$, then the length of $$BF$$ is          ."},{"type":"image_path","image_path":"images/340_q0.png"}],"answer":"$$\\frac{75}{41}$$ or $$\\frac{75}{17}$$"}
{"id":"343","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle A B C = 90 \\circ$$, connect $$B D$$, $$A B = A D = 2$$, $$B D = 2 \\sqrt{2}$$, points $$E$$, $$F$$ lie on sides $$B C$$, $$C D$$ respectively, and $$D F = C E$$, connect $$B F$$, $$D E$$, if $$C D = 3$$, then the minimum value of $$B F + D E$$ is         ."},{"type":"image_path","image_path":"images/343_q0.png"}],"answer":"$$\\sqrt{29}$$"}
{"id":"346","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A D = 17$$, $$A B = 16$$, point E is a moving point on the sides of the rectangle. Connect $$D E$$, fold $$\\triangle A D E$$ along the line $$D E$$, and point A lands at point F (point F is below the line $$A D$$). Connect $$C F$$. When $$\\triangle C D F$$ is an isosceles triangle with $$D F$$ as the leg, the value of $$\\tan \\angle A D E$$ is      ."},{"type":"image_path","image_path":"images/346_q0.png"}],"answer":"4 or $$\\frac{1}{4}$$"}
{"id":"352","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A D = 2$$, $$A B = 4$$, $$E$$ and $$F$$ are moving points on sides $$A B$$ and $$C D$$ respectively, and $$E F \\bot A C$$. Then the minimum value of $$A F + C E$$ is          ."},{"type":"image_path","image_path":"images/352_q0.png"}],"answer":"5"}
{"id":"360","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the radius of ⊙O is 1, point B is the midpoint of radius OA, and point C is a moving point on ⊙O. Rotate CB clockwise by 90° around point B to obtain BD. Let M be the midpoint of AD. Then the maximum value of OM is ______."},{"type":"image_path","image_path":"images/360_q0.png"}],"answer":"$$\\frac{1}{2} + \\frac{\\sqrt{10}}{4}$$"}
{"id":"369","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ and $$\\triangle A D E$$ are both equilateral triangles. Connect $$C E$$ and $$B E$$."},{"type":"image_path","image_path":"images/369_q0.png"},{"type":"text","text":"(1) If $$A E = 4$$, then the area of $$\\triangle A D E$$ is      ; (2) If $$B C \\bot C E$$ and $$B E = 4$$, then the maximum area of $$\\triangle A D E$$ is      ."}],"answer":"$$4 \\sqrt{3}$$;     $$6 + 4 \\sqrt{3}$$/$$4 \\sqrt{3} + 6$$."}
{"id":"370","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, given $$\\angle M C N = 30 \\circ$$, if points $$E$$ and $$A$$ lie on ray $$C M$$, and satisfy $$A C = 6 \\sqrt{3}$$, $$E C : A E = 2 : 1$$, and $$G$$ is a moving point on ray $$C N$$, simultaneously construct $$\\angle G E H = 30^\\circ$$ on the right side of $$E G$$, and satisfy $$E G = \\sqrt{3} E H$$, then the area of $$\\triangle C E H$$ is     . If the trajectory of point $$H$$ intersects ray $$C N$$ at point $$Q$$, when $$2 A H + H Q$$ attains its minimum value, the value of $$A H$$ is     ."},{"type":"image_path","image_path":"images/370_q0.png"}],"answer":"$$4 \\sqrt{3}$$     $$\\frac{4}{3} \\sqrt{3}$$"}
{"id":"371","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A B C = 30 \\circ$$, an equilateral triangle $$\\triangle D C B$$ is constructed on side $$B C$$ above it. Points $$E$$ and $$F$$ are two moving points on sides $$A C$$ and $$A B$$ respectively, such that $$A E = B F$$. If $$B C = 6$$, then the minimum value of $$D E + D F$$ is           ."},{"type":"image_path","image_path":"images/371_q0.png"}],"answer":"$$2 \\sqrt{21}$$"}
{"id":"372","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the diameter $$AB$$ of $$\\bigodot O$$ intersects the chord $$CD$$ at point E, and point P is a point on $$CD$$ such that $$\\angle APB = 120^\\circ$$. If $$AB = 6$$ and $$CD = 4$$, then the maximum value of $$PA \\cdot PB$$ is     , and the maximum value of $$PA + PB$$ is        ."},{"type":"image_path","image_path":"images/372_q0.png"}],"answer":"8     $$2 \\sqrt{11}$$"}
{"id":"374","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$BC$$ is the diameter of $$\\bigodot O$$, point A lies on $$\\bigodot O$$, $$AD \\bot BC$$, with foot at D, $$\\overset{⌢}{AE} = \\overset{⌢}{AB}$$, the extensions of $$BE$$ and $$AC$$ intersect at point G, the extension of $$AD$$ intersects $$BE$$ at point F. If $$BG = 10$$ and $$BD - DF = 1$$, then $$AB =$$        ."},{"type":"image_path","image_path":"images/374_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"376","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle B = 120 \\circ$$, point E is a point on $$B C$$, connect $$A E$$, rotate $$A E$$ clockwise around point E by $$120 \\circ$$ to $$F E$$, connect $$A F$$ intersecting $$C D$$ at point G, if $$\\frac{D G}{C G} = \\frac{1}{4}$$, then $$\\frac{B E}{C E} =$$      ."},{"type":"image_path","image_path":"images/376_q0.png"}],"answer":"$$\\frac{4}{3}$$/$$1 \\frac{1}{3}$$"}
{"id":"381","difficulty":"0.2","question_list":[{"type":"text","text":"Given rectangle $$A B C D$$, connect diagonal $$A C$$, point $$F$$ lies on side $$C D$$, connect $$A F$$, point $$E$$ is the midpoint of $$A F$$, connect $$D E$$, if $$\\angle D E F = \\angle B A C, D E = \\sqrt{5}, C F = 1$$, then the length of $$A B$$ is      ."},{"type":"image_path","image_path":"images/381_q0.png"}],"answer":"3"}
{"id":"388","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, and E is a point on side $$C D$$. Connect $$A E$$, and from point B, draw $$B F \\bot A E$$ intersecting at point F. Point G is the reflection of point F across line $$C D$$, and H is the midpoint of $$C G$$. Then the minimum value of $$A H$$ is            ."},{"type":"image_path","image_path":"images/388_q0.png"}],"answer":"$$3 \\sqrt{5} - 1$$/$$- 1 + 3 \\sqrt{5}$$"}
{"id":"389","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = \\sqrt{5}$$, $$B C = 2$$, point $$D$$ is any moving point on $$B C$$ (not coinciding with $$B$$ or $$C$$). From point $$B$$, draw $$B H \\bot A D$$, with foot at point $$H$$, and connect $$C H$$. Then the minimum value of $$C H$$ is        ."},{"type":"image_path","image_path":"images/389_q0.png"}],"answer":"$$\\frac{\\sqrt{13} - \\sqrt{5}}{2}$$"}
{"id":"392","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle B A C = 120 \\circ$$, point $$D$$ is a point on side $$A B$$ (not coinciding with point $$B$$), connect $$C D$$, rotate segment $$C D$$ counterclockwise by $$90 \\circ$$ about point $$D$$, and let the image of point $$C$$ be point $$E$$, connect $$B E$$. If $$A B = 10$$, then the maximum area of $$\\triangle B D E$$ is       ."},{"type":"image_path","image_path":"images/392_q0.png"}],"answer":"$$\\frac{225}{8}$$"}
{"id":"395","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$, point E is the midpoint of side $$B C$$, connect $$A E$$, points F and G lie on $$A E$$ and $$D E$$ respectively, and $$F G \\parallel A D$$, the reflection of point E over $$F G$$ is $$E^{'}$$, $$A D$$ intersects $$E^{'} F$$ and $$E^{'} G$$ at H and I respectively. If $$A H + D I = H I$$, then the area of quadrilateral $$F E G E^{'}$$ is                    ."},{"type":"image_path","image_path":"images/395_q0.png"}],"answer":"$$\\frac{27}{2}$$"}
{"id":"398","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$A D = 10$$. A moving point $$E$$ lies on side $$A B$$, and an arc is drawn with center at point $$E$$ and radius $$B E$$. Point $$G$$ is a moving point on this arc. If $$A E = 1$$, connect $$C G$$ and $$D G$$, and let $$P$$ and $$Q$$ be the midpoints of $$D G$$ and $$C G$$, respectively. Connect $$P Q$$, and let $$M$$ be the midpoint of $$P Q$$. Then the minimum value of $$C M$$ is      ."},{"type":"image_path","image_path":"images/398_q0.png"}],"answer":"$$\\frac{2 \\sqrt{41} - 5}{2}$$"}
{"id":"403","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A C = 6$$, $$B C = 8$$. Point $$D$$ is the midpoint of the hypotenuse $$A B$$, and point $$P$$ is a moving point on side $$A C$$. Connect $$P D$$, and rotate segment $$P D$$ clockwise by $$90 \\circ$$ about point $$D$$ to obtain segment $$P^{'} D$$. Segment $$P^{'} D$$ intersects side $$B C$$ at point $$Q$$. When $$\\triangle B Q P^{'}$$ is a right triangle, the length of $$A P$$ is      ."},{"type":"image_path","image_path":"images/403_q0.png"}],"answer":"3 or 5/5 or 3"}
{"id":"408","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, there is a moving point E on side BC of rectangle ABCD. Connect AE, and construct rectangle AEGF with AE as one side, such that side FG passes through point D. If AB = √3 and BC = 4, when triangle AED is an isosceles triangle, the length of BE is        ."},{"type":"image_path","image_path":"images/408_q0.png"}],"answer":"2 or $$\\sqrt{13}$$ or $$4 - \\sqrt{13}$$"}
{"id":"409","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C A B = 90 \\circ$$. Rotate side $$C A$$ counterclockwise about point $$C$$ by $$90 \\circ$$ to obtain segment $$C E$$, and rotate side $$C B$$ clockwise about point $$B$$ by $$90 \\circ$$ to obtain segment $$B D$$. Let $$C D$$ intersect $$A E$$ at point $$F$$. If $$E F = \\sqrt{2}$$ and $$D F = \\sqrt{10}$$, then the length of side $$A B$$ is          ."},{"type":"image_path","image_path":"images/409_q0.png"}],"answer":"2"}
{"id":"413","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, there is a rectangular paper $$A B C D$$, with $$A B = 7$$, $$B C = 3$$. Points $$M$$ and $$N$$ lie on sides $$A B$$ and $$C D$$ respectively, and $$C N = 1$$. The quadrilateral $$B C M N$$ is folded along $$M N$$, causing points $$B$$ and $$C$$ to land on points $$B^{'}$$ and $$C^{'}$$ respectively. As point $$M$$ moves from point $$A$$ to point $$B$$, if side $$M B^{'}$$ intersects side $$C D$$ at point $$E$$, then the length of the path traced by point $$E$$ is      ."},{"type":"image_path","image_path":"images/413_q0.png"}],"answer":"$$\\sqrt{10} - \\frac{9}{4}$$"}
{"id":"415","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, $$D$$ is the midpoint of side $$B C$$, point $$E$$ lies on side $$A C$$, connect $$D E$$, fold $$\\triangle C D E$$ along $$D E$$ by $$180^\\circ$$ to obtain $$\\triangle F D E$$, connect $$B E$$ and $$B F$$, given that $$A C = \\sqrt{10} B F$$, $$S_{\\triangle B E F} = \\frac{3}{2}$$, then the area of $$\\triangle A B C$$ is      ."},{"type":"image_path","image_path":"images/415_q0.png"}],"answer":"$$5 \\sqrt{3}$$"}
{"id":"417","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, $$AB = 4$$, $$BC = 3$$, translate $$\\triangle BCD$$ along ray BD by length $$a \\left(\\right. a > 0 \\left.\\right)$$ to obtain $$\\triangle B^{'} C^{'} D^{'}$$, connect $$AB^{'}$$, $$AD^{'}$$, then when $$\\triangle AB^{'} D^{'}$$ is a right triangle, the length of $$a$$ is           ."},{"type":"image_path","image_path":"images/417_q0.png"}],"answer":"$$\\frac{7}{5}$$ or $$\\frac{16}{5}$$"}
{"id":"418","difficulty":"0.2","question_list":[{"type":"text","text":"In $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C$$, point $$D$$ is a point on side $$A B$$, through point $$D$$ draw $$D E \\bot A B$$, intersecting $$B C$$ at point $$E$$, connect $$A E$$, take the midpoint $$P$$ of $$A E$$, connect $$D P$$, $$C P$$. If $$B C = 3 B D = 6 \\sqrt{2}$$, rotate $$\\triangle B D E$$ around point $$B$$ freely in the plane, when $$B E \\bot A B$$, then the length of segment $$C P =$$        ."},{"type":"image_path","image_path":"images/418_q0.png"}],"answer":"$$8$$ or $$4$$"}
{"id":"419","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$E$$ is the midpoint of $$AB$$. Rotate $$DE$$ clockwise around point $$E$$ by $$90^\\circ$$ to obtain $$EF$$. Connect $$CF$$ and $$AF$$, and let $$AF$$ intersect $$DE$$ at point $$M$$. If $$CF = 6$$, then the length of $$DM$$ is        ."},{"type":"image_path","image_path":"images/419_q0.png"}],"answer":"$$\\frac{18 \\sqrt{10}}{7}$$/$$\\frac{18}{7} \\sqrt{10}$$"}
{"id":"421","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point M is a point on side $$CD$$. Connect $$AM$$. Rotate $$\\triangle ADM$$ clockwise about point $$A$$ by $$90^\\circ$$ to obtain $$\\triangle ABN$$. On $$AM$$ and $$AN$$, respectively, take points $$E$$ and $$F$$ such that $$AE = AF = BC$$. Connect $$EF$$, intersecting diagonal $$BD$$ at point $$G$$. Connect $$AG$$ and extend it to intersect $$BC$$ at point $$H$$. If $$AM = \\frac{25}{3}$$ and $$CH = 2$$, then the length of $$AG$$ is ________."},{"type":"image_path","image_path":"images/421_q0.png"}],"answer":"$$\\frac{40}{7}$$/$$5 \\frac{5}{7}$$"}
{"id":"422","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points P, M, N move along sides $$A B$$, $$A D$$, $$B C$$ respectively, and segment $$M N$$ always passes through the center of symmetry of the rectangle. The minimum perimeter of $$\\triangle P M N$$ is            ."},{"type":"image_path","image_path":"images/422_q0.png"}],"answer":"$$2 \\sqrt{13} + 4$$"}
{"id":"425","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$A B = A C = 8$$, $$B O = \\frac{1}{4} A B$$, point $$M$$ is a moving point on side $$B C$$, rotate segment $$O M$$ counterclockwise around point $$O$$ by $$90^\\circ$$ to obtain $$O N$$, and connect $$A N$$ and $$C N$$. (1) When point $$N$$ lies on $$A B$$, $$A N =$$     ; (2) The minimum perimeter of $$\\triangle C A N$$ is                    ."},{"type":"image_path","image_path":"images/425_q0.png"}],"answer":"4     $$8 + 4 \\sqrt{10}$$"}
{"id":"429","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, it is given that quadrilateral $$A B C D$$ is a rhombus, $$A B = 4$$, $$\\angle C = 60 \\circ$$, $$B D$$ is a diagonal, and E is a moving point on side $$C D$$, with $$E F \\parallel B D$$ intersecting $$B C$$ at point F. Connect $$A E$$ and $$A F$$, and let G be the midpoint of $$A E$$. Connect $$F G$$.\n① If E is the midpoint of $$D C$$, then the length of $$C F$$ is     ;\n② As point E moves, the minimum value of $$G F$$ is       ."},{"type":"image_path","image_path":"images/429_q0.png"}],"answer":"$$2$$     $$3$$"}
{"id":"433","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A B = 6$$, $$E$$ is a trisection point of side $$C D$$, $$\\angle D = 60^\\circ$$, fold $$\\triangle A D E$$ along $$A E$$ to obtain $$\\triangle A F E$$, and let line $$E F$$ intersect $$B C$$ at point $$P$$. Then $$P C =$$       ."},{"type":"image_path","image_path":"images/433_q0.png"}],"answer":"$$\\frac{3}{2}$$ or $$\\frac{6}{5}$$"}
{"id":"436","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 30 \\circ$$, $$A B = A C = 2$$, point E is a moving point on ray $$A C$$, $$D E \\parallel A B$$, and $$D E = 2$$. When the value of $$A D + B D$$ is minimized, the measure of $$\\angle D B C$$ is     ."},{"type":"image_path","image_path":"images/436_q0.png"}],"answer":"$$45 \\circ$$"}
{"id":"439","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 45 \\circ$$, $$C D \\bot A B$$ at point $$D$$, $$A E \\bot B C$$ at point $$E$$, $$A E$$ intersects $$C D$$ at point $$F$$, connect $$B F$$ and $$D E$$, among the following conclusions: ① $$A F = B C$$; ② $$2 \\cos \\angle D E B = \\frac{1}{2}$$, ③ $$A E - C E = \\sqrt{2} E D$$, ④ if $$\\angle C A E = 30 \\circ$$, then $$\\frac{A F + B F}{A C} = 1$$, the correct ones are      . (fill in the serial numbers)."},{"type":"image_path","image_path":"images/439_q0.png"}],"answer":"①③④"}
{"id":"441","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4\\sqrt{2}$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Point $$E$$ is a point on diagonal $$AC$$, connect $$BE$$, draw $$EF \\bot BE$$, intersecting $$BD$$ and $$CD$$ at points $$F$$ and $$G$$ respectively, connect $$BG$$ intersecting $$AC$$ at point $$H$$. Fold $$\\triangle EGH$$ along $$EG$$, and the image point $$P$$ of point $$H$$ falls exactly on $$BD$$, forming $$\\triangle EPG$$. If point $$G$$ is the midpoint of $$CD$$, then the area of $$\\triangle PFG$$ is        ."},{"type":"image_path","image_path":"images/441_q0.png"}],"answer":"$$\\frac{5}{3}$$/$$1 \\frac{2}{3}$$"}
{"id":"444","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A D \\parallel B C$$, $$A C$$ and $$B D$$ are diagonals, $$A C \\bot C D$$, $$A B = A C$$. If $$\\angle A B D = 2 \\angle A D C$$ and $$C D = 2 \\sqrt{5}$$, then the length of $$A D$$ is            ."},{"type":"image_path","image_path":"images/444_q0.png"}],"answer":"$$2 \\sqrt{6}$$"}
{"id":"445","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, diagonals AB and CD intersect at point E, $$CA = CB = CD$$, $$\\angle ACB = 90^\\circ$$, if $$S_{\\triangle ACE} - S_{\\triangle BDE} = 8$$, then the length of segment AD is         ."},{"type":"image_path","image_path":"images/445_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"446","difficulty":"0.2","question_list":[{"type":"text","text":"Given in rectangle $$A B C D$$, $$A D = 9$$, $$A B = 12$$, O is the center of the rectangle; in $$\\text{Rt} \\triangle A E F$$, $$\\angle E A F = 90 \\circ$$, $$A E = 6$$, $$A F = 8$$. Rotate $$\\triangle A E F$$ around point A in a clockwise direction for one full turn, then the height from side $$E F$$ is      . Connect $$C E$$, take the midpoint M of $$C E$$, connect $$F M$$, write the range of values for $$F M$$      ."},{"type":"image_path","image_path":"images/446_q0.png"}],"answer":"$$4.8$$     $$\\frac{2 \\sqrt{73} - 15}{2} \\leq F M \\leq \\frac{15 + 2 \\sqrt{73}}{2}$$"}
{"id":"449","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$ with side length 4, point E is a moving point on side $$BC$$ (point E does not coincide with B or C). Connect $$AE$$. From point B, draw $$BF \\bot AE$$ intersecting at point F. Point G is the reflection of point C across line $$BF$$. Connect $$AG$$, $$DG$$, and $$GF$$. When $$GF$$ attains its minimum value, the area of $$\\triangle AGD$$ is            ."},{"type":"image_path","image_path":"images/449_q0.png"}],"answer":"$$8 - \\frac{16 \\sqrt{5}}{5}$$"}
{"id":"452","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$A B C$$, $$A D = B E = C F$$, $$A N = C M = B P$$, $$D E = 7$$, $$M N = 5$$, and $$M N \\parallel D E$$, then the length of $$A B$$ is          ."},{"type":"image_path","image_path":"images/452_q0.png"}],"answer":"$$\\frac{35 \\sqrt{13}}{13}$$"}
{"id":"455","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C$$, $$\\angle A C B = 90 \\circ$$, point D lies on side $$A C$$, $$A D = 2 C D$$, an equilateral triangle $$\\triangle B D E$$ is constructed with $$B D$$ as a side, and $$D E$$ intersects $$A B$$ at point F. Then $$\\frac{E F}{F D} =$$      ."},{"type":"image_path","image_path":"images/455_q0.png"}],"answer":"$$\\frac{2 \\sqrt{3} - 1}{2}$$"}
{"id":"456","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a square with side length $$1 \\text{cm}$$, point F is a moving point on the extension of side $$B C$$, point E is above line $$B C$$, and $$E F = B F$$, $$E F \\bot B F$$. If $$\\triangle D E F$$ is exactly an isosceles triangle, then the length of $$C F$$ is      $$\\text{cm}$$."},{"type":"image_path","image_path":"images/456_q0.png"}],"answer":"$$\\sqrt{2} + 1$$ or $$1$$"}
{"id":"459","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, point D is a moving point on the side BC of the equilateral triangle ABC, with AB = 2. Construct an equilateral triangle ADE on the right side of AD. AC intersects DE at point F. Which of the following conclusions are correct? (Fill in the serial numbers) ① ∠ACD = ∠ACE; ② AF · CF = DF · EF; ③ The maximum area of triangle CDE is √3/2; ④ The minimum length of segment DE is √3."},{"type":"image_path","image_path":"images/459_q0.png"}],"answer":"①②④"}
{"id":"471","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = \\sqrt{7}$$, $$B C = 2 \\sqrt{3}$$, $$P$$ is a point inside $$\\triangle A B C$$, $$\\angle A P B = \\angle A P C = 120^\\circ$$, then the value of $$P A + P B + P C$$ is      ."},{"type":"image_path","image_path":"images/471_q0.png"}],"answer":"5"}
{"id":"475","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 8$$, point $$P$$ is any point on ray $$BC$$ (not coinciding with points $$B$$ and $$C$$). Connect $$AP$$, and construct square $$APGH$$ on the right side of $$AP$$. Connect $$AG$$, intersecting ray $$CD$$ at point $$E$$. When the length of $$ED$$ is 2, the length of $$BP$$ is        ."},{"type":"image_path","image_path":"images/475_q0.png"}],"answer":"$$\\frac{24}{5}  or  \\frac{40}{3}$$"}
{"id":"476","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$B D$$ is a diagonal, and $$E$$ is a point on side $$A D$$. Connect $$C E$$ intersecting $$B D$$ at point $$O$$. If $$\\angle A = \\angle B C D = \\angle B O C = 120 \\circ$$, $$A D = \\frac{69}{2}$$, $$A B = 12$$, $$\\frac{B C}{C D} = \\frac{4}{3}$$, then the value of $$\\frac{B D}{C E}$$ is      ."},{"type":"image_path","image_path":"images/476_q0.png"}],"answer":"$$\\frac{23}{12}$$/$$1 \\frac{11}{12}$$"}
{"id":"479","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$A B = 8$$, point P is a moving point on side $$A B$$, and a circle with diameter $$B P$$ intersects $$C P$$ at point Q. If the minimum length of segment $$A Q$$ is 4, then the area of $$\\triangle A B C$$ is     ."},{"type":"image_path","image_path":"images/479_q0.png"}],"answer":"48"}
{"id":"485","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$CD$$ is the median on the hypotenuse $$AB$$ of $$\\text{Rt} \\triangle ABC$$, point $$E$$ is the midpoint of $$CD$$, connect $$AE$$, point $$F$$ is the midpoint of $$AE$$, connect $$DF$$, if $$DE = DF = 2$$, then the length of segment $$AC$$ is           ."},{"type":"image_path","image_path":"images/485_q0.png"}],"answer":"$$2 \\sqrt{10}$$ or $$2 \\sqrt{6}$$/$$2 \\sqrt{6}$$ or $$2 \\sqrt{10}$$"}
{"id":"486","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$AC = AB$$, $$\\angle BAC = 90^\\circ$$, $$D$$ is a point on side $$AC$$, connect $$BD$$, $$AF \\perp BD$$ at point $$F$$, point $$E$$ lies on $$BF$$, connect $$AE$$ and $$CE$$, $$\\angle EAF = 45^\\circ$$, if $$\\tan \\angle ECD = \\frac{3}{4}$$, $$BC = 6$$, then the length of $$BE$$ is  ."},{"type":"image_path","image_path":"images/486_q0.png"}],"answer":"$$\\frac{3 \\sqrt{10}}{5}$$"}
{"id":"489","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the diagonals of rhombus $$A B C D$$ intersect at point O. Points M and N are moving points on segments $$A B$$ and $$A C$$, respectively, such that $$A M = C N$$. If $$A B = 5$$ and $$B D = 6$$, then the minimum value of $$D M + D N$$ is      ."},{"type":"image_path","image_path":"images/489_q0.png"}],"answer":"$$\\frac{13 \\sqrt{10}}{5}$$"}
{"id":"491","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an isosceles right triangle, D is the midpoint of $$A B$$, E and F are points on $$A C$$ and $$B C$$ respectively such that $$D F \\bot D E$$, given $$A E = 2$$, $$C E = 5$$, M is a point on $$B C$$, connect $$M E$$, and satisfies $$\\angle C M E = 2 \\angle A D E$$, then $$E M = \\\\$$           ."},{"type":"image_path","image_path":"images/491_q0.png"}],"answer":"$$\\frac{29}{4}$$"}
{"id":"492","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in right triangle $$A B C$$, $$\\angle A B C = 90 \\circ, A B = 8, B C = 6$$, the angle bisectors of $$\\angle C A B$$ and $$\\angle A C B$$, namely $$A E, C D$$, intersect at point P, the altitude from $$B$$ to side $$A C$$, denoted as $$B F$$, intersects $$A E$$ and $$C D$$ at points G and H respectively, M and N are the midpoints of $$D H$$ and $$E G$$ respectively, connect $$M N$$, $$B M$$, $$B N$$, which of the following statements is correct                    .\n①$$B F = 4.8$$, ② the ratio of the areas of $$\\triangle A B P$$ and $$\\triangle C B P$$ is $$3 : 4$$, ③ $$\\triangle B D H$$ is an isosceles triangle, ④ $$B N \\bot A E$$, ⑤ $$\\angle M N P = \\angle E A B$$ (please fill in the corresponding serial numbers)"},{"type":"image_path","image_path":"images/492_q0.png"}],"answer":"①③④⑤"}
{"id":"497","difficulty":"0.2","question_list":[{"type":"text","text":"Given that the vertices M, N, P, Q of rectangle $$M N P Q$$ lie respectively on the sides $$D E$$, $$F A$$, $$A B$$, $$C D$$ of regular hexagon $$A B C D E F$$, and as point M moves from E to D, the following judgments about rectangle $$M N P Q$$:\n① The area and perimeter of rectangle $$M N P Q$$ remain constant;\n② The area of rectangle $$M N P Q$$ gradually decreases;\n③ The perimeter of rectangle $$M N P Q$$ gradually increases;\n④ The length of the diagonal of rectangle $$M N P Q$$ has a minimum value.\nThe correct one(s) is/are    . (Fill in the serial number)"},{"type":"image_path","image_path":"images/497_q0.png"}],"answer":"②④/④②"}
{"id":"501","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point D is a point on $$B C$$, $$\\angle A D C = 60 \\circ$$, point E lies on segment $$A D$$, $$\\angle B E C = 120 \\circ$$, if $$B C = 3 \\sqrt{3}$$, $$A E = 2 \\sqrt{3}$$, then the maximum value of $$A C$$ is        ."},{"type":"image_path","image_path":"images/501_q0.png"}],"answer":"$$3 + \\sqrt{3}$$/$$\\sqrt{3} + 3$$"}
{"id":"503","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the square $$A B C D$$ has a side length of 6, point E is the midpoint of $$B C$$, connecting $$A E$$ intersects the diagonal $$B D$$ at point G, connecting $$C G$$ and extending it intersects $$A B$$ at point F, connecting $$D E$$ intersects $$C F$$ at point H, and connecting $$A H$$. The following conclusions: ① $$C F \\bot D E$$; ② $$\\frac{C H}{H F} = \\frac{2}{3}$$; ③ $$A D = A H$$; ④ $$G H = \\frac{3}{5} \\sqrt{5}$$, among which the correct conclusions are      ."},{"type":"image_path","image_path":"images/503_q0.png"}],"answer":"①②③"}
{"id":"508","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$\\text{Rt} \\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, points D and E lie on segments $$A B$$ and $$A C$$ respectively, with $$A D = A E$$; BE and CD intersect at point N; AF ⊥ BE intersects BC at point F; FG ⊥ CD intersects AC at point M and intersects the extension of BE at point G. The following statements: ① $$\\angle A B E = \\angle F A C$$; ② $$G E = M E$$; ③ $$B G = A F + F G$$; ④ $$C_{\\triangle A F M} = B E + C M$$; ⑤ $$S_{\\triangle B D N} : S_{\\triangle A F C} = C E : A C$$. Among these, the correct ones are      ."},{"type":"image_path","image_path":"images/508_q0.png"}],"answer":"①③⑤"}
{"id":"517","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, point $$E$$ is a moving point on $$A C$$, connect $$D E$$, draw line $$l$$ perpendicular to $$A C$$ passing through point $$C$$, draw $$D F \\bot D E$$ intersecting $$l$$ at point $$F$$, draw $$D G \\bot E F$$ at point $$G$$, $$t a n \\angle E D G = \\sqrt{2}$$, point $$H$$ is the midpoint of $$A D$$, connect $$H E$$, then the minimum value of $$H E + \\frac{\\sqrt{6}}{3} E C$$ is   ."},{"type":"image_path","image_path":"images/517_q0.png"}],"answer":"$$\\frac{5 \\sqrt{2}}{3}$$/$$\\frac{5}{3} \\sqrt{2}$$"}
{"id":"518","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, point E is connected to B and C. From C, draw CF perpendicular to CE, intersecting the extension of BE at point F. Connect DF and DE. If CE = CF = 1, DE = √6, which of the following conclusions are correct: ① △CBE ≈ △CDF; ② BF ⊥ DF; ③ The distance from point D to CF is 2; ④ S_{DECF} = √2 + 1. The correct conclusions are ______. (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/518_q0.png"}],"answer":"$$\\textcircled{1}\\textcircled{2}$$/$$\\textcircled{2}\\textcircled{1}$$"}
{"id":"525","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, sector $$A O B$$, with $$O B = \\sqrt{2}$$, $$\\angle A O B = 90 \\circ$$, $$C$$ is any point on arc $$A B$$, draw $$C D \\bot O B$$ at point $$D$$, let the incenter of $$\\triangle O D C$$ be $$E$$, connect $$O E$$ and $$C E$$, when point $$C$$ moves from point $$B$$ to point $$A$$, the path length traced by the incenter $$E$$ is           ."},{"type":"image_path","image_path":"images/525_q0.png"}],"answer":"$$\\frac{\\pi}{2}$$/$$\\frac{1}{2} \\pi$$/$$0.5 \\pi$$"}
{"id":"527","difficulty":"0.2","question_list":[{"type":"text","text":"In $$R t \\triangle A B C$$, $$D$$ is the midpoint of the hypotenuse, $$E$$ and $$F$$ lie on side $$B C$$, $$D E \\bot A F$$, $$\\angle A E B = 2 \\angle C E D = 2 \\angle C A F$$, $$C F = 4$$, then $$B E =$$        ; $$E F =$$          ."},{"type":"image_path","image_path":"images/527_q0.png"}],"answer":"$$1$$     $$\\frac{\\sqrt{17} - 3}{2}$$"}
{"id":"531","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C = 5 \\text{cm}$$; in $$\\triangle C D E$$, $$\\angle D C E = 90 \\circ$$, $$D C = E C = 3 \\text{cm}$$; the lines $$B D$$ and $$A E$$ intersect at point F. Now rotate $$\\triangle D C E$$ around point C for one full revolution. During the rotation, $$\\angle F =$$        $$\\circ$$, and the maximum length of segment $$A F$$ is      $$\\text{cm}$$."},{"type":"image_path","image_path":"images/531_q0.png"}],"answer":"90     7"}
{"id":"536","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A B =\\text{2} \\sqrt{\\text{3}}$$, diagonal $$A C =\\text{2}$$, $$\\angle B A C =\\angle A C D =\\text{60}°$$, let $$A D = k \\cdot B D$$, then the minimum value of $$k$$ is            ."},{"type":"image_path","image_path":"images/536_q0.png"}],"answer":"$$\\sqrt{\\text{2}} −\\text{1}$$/$$- 1 + \\sqrt{2}$$"}
{"id":"539","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, $$\\angle BCD = \\text{90}°$$, $$BC = \\sqrt{\\text{3}} CD$$, $$\\angle BAC = \\text{60}°$$, if $$AB = \\text{5}$$, $$AD = \\text{2}$$, then the length of segment AC is      ."},{"type":"image_path","image_path":"images/539_q0.png"}],"answer":"$$\\text{2}.\\text{5}+ \\sqrt{\\text{3}}$$"}
{"id":"544","difficulty":"0.2","question_list":[{"type":"text","text":"Given square ABCD, AB=6, DP bisects ∠ADC and DP=$$\\sqrt{2}$$, AE=2, connect EP and CP, point F is on EP such that EF:FP=11:3. Connect CF, point M is on EC such that ∠EFM=∠CFP, draw MN⊥CF intersecting CP at point N, then PN=          ."},{"type":"image_path","image_path":"images/544_q0.png"}],"answer":"$$\\frac{4 \\sqrt{26}}{7}$$/$$\\frac{4}{7} \\sqrt{26}$$"}
{"id":"552","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠BAC = 60°, its perimeter is 20, ⊙I is the incircle of △ABC, with radius $$\\sqrt{3}$$, then the diameter of the circumcircle of △BIC is                   ."},{"type":"image_path","image_path":"images/552_q0.png"}],"answer":"$$\\frac{14 \\sqrt{3}}{3}$$"}
{"id":"554","difficulty":"0.2","question_list":[{"type":"text","text":"In Rt△ABC, AB = AC = $$4 \\sqrt{2}$$, BO = $$\\frac{1}{4}$$AB, point M is a moving point on side BC, rotate segment OM counterclockwise around point O by 90° to ON, connect AN and CN, then the minimum perimeter of △CAN is             ."},{"type":"image_path","image_path":"images/554_q0.png"}],"answer":"$$4 \\sqrt{2} + 4 \\sqrt{5}$$"}
{"id":"562","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠BAC = 30°, and AB = AC. P is a point inside △ABC. If the minimum value of AP + BP + CP is 4$$\\sqrt{2}$$, then the length of BC is         ."},{"type":"image_path","image_path":"images/562_q0.png"}],"answer":"$$2 \\sqrt{6} - 2 \\sqrt{2}$$"}
{"id":"563","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, points E and F are two moving points on sides $$AB$$ and $$BC$$, respectively, and the perimeter of square $$ABCD$$ is twice the perimeter of $$\\triangle BEF$$. Connect $$DE$$ and $$DF$$, which intersect diagonal $$AC$$ at points M and N, respectively. The following conclusions are given: ① If $$AE = 2$$, $$CF = 3$$, then $$EF = 4$$; ② $$\\angle EFN + \\angle EMN = 180^\\circ$$; ③ If $$AM = 2$$, $$CN = 3$$, then $$MN = 4$$; ④ If $$\\frac{MN}{AM} = 2$$, $$BE = 3$$, then $$EF = 4$$. The correct conclusion(s) is/are            ."},{"type":"image_path","image_path":"images/563_q0.png"}],"answer":"②"}
{"id":"564","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta ABC$$, $$\\angle C = 90^{\\circ}$$, $$AC = 8$$, $$BC = 6$$, O is the incenter, and a line passing through point O intersects AC and AB at points D and E, respectively. If $$DE = CD + BE$$, then the length of segment CD is          ."},{"type":"image_path","image_path":"images/564_q0.png"}],"answer":"2 or $$\\frac{1}{2}$$/$$\\frac{1}{2}$$ or 2"}
{"id":"569","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C > B C$$, three squares $$A B H L$$, $$A C D E$$, $$B C F G$$ are constructed outwardly on the three sides of $$\\triangle A B C$$, and $$D F$$ is connected. From point $$C$$, draw a perpendicular $$C J$$ to $$A B$$, with foot at $$J$$, intersecting $$D F$$ and $$L H$$ at points $$I$$ and $$K$$, respectively. If $$C I = 5$$, $$C J = 4$$, then the area of quadrilateral $$A J K L$$ is         ."},{"type":"image_path","image_path":"images/569_q0.png"}],"answer":"80"}
{"id":"581","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, $$\\angle B = \\angle C = 90^\\circ$$, point E lies on side BC, and AE is connected. If $$\\angle BAD - \\frac{1}{2} \\angle BAE = 45^\\circ$$, $$AB = BC = 4CD$$, $$AE = 3$$, then the length of segment AD is      ."},{"type":"image_path","image_path":"images/581_q0.png"}],"answer":"$$\\frac{18}{5}$$"}
{"id":"584","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in an equilateral triangle ABC with side length 2, P0 is the midpoint of side BC. A beam of light emanates from P0, strikes point P1 on AC, then reflects successively to points P2 and P3 on AB and BC, respectively, with 1 < BP3 < $$\\frac{3}{2}$$ (angle of reflection equals angle of incidence). The range of values for P1C is     ."},{"type":"image_path","image_path":"images/584_q0.png"}],"answer":"$$1 < P_{1} C < \\frac{7}{6}$$"}
{"id":"589","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, a circle ⊙O with radius 2 is tangent to line l at point A. P is a moving point on ⊙O (not coinciding with point A). From point P, draw PB ⊥ l, with foot at B, and connect PA. Let PA = x and PB = y. Then the maximum value of (x - y) is        ."},{"type":"image_path","image_path":"images/589_q0.png"}],"answer":"1"}
{"id":"592","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, $$\\angle A B C = \\angle A C B$$, point O is the circumcenter of △ABC. Connect CO and extend it to intersect side AB at point P, $$A P = 3$$, $$B P = 4$$, then the value of $$c o s \\angle A B C$$ is        ."},{"type":"image_path","image_path":"images/592_q0.png"}],"answer":"$$\\frac{\\sqrt{6}}{6}$$/$$\\frac{1}{6} \\sqrt{6}$$"}
{"id":"596","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, $$\\angle BCA = 90^{\\circ}$$, $$AC = 6$$, $$BC = 8$$, point D is the midpoint of AB, and point E is a moving point on BC. Triangle △BDE is folded along DE to obtain △FDE, and EF intersects CD at point M. When △DFM is a right triangle, the length of segment BE is   ."},{"type":"image_path","image_path":"images/596_q0.png"}],"answer":"$$\\frac{11}{2}$$ or 7"}
{"id":"603","difficulty":"0.2","question_list":[{"type":"text","text":"Given: As shown in the figure, in rectangle ABCD, AB = 6, BC = 9. Point E is a point on diagonal AC. The circle ⊙O passing through points C, D, and E intersects AD and BC at points F and G respectively. Connect ED, EF, and EG. Extend GE to intersect AD at point H. If △HEF is an isosceles triangle, then the length of CE is          ."},{"type":"image_path","image_path":"images/603_q0.png"}],"answer":"6 or $$\\frac{3}{2} \\sqrt{13}$$ or $$\\frac{24 \\sqrt{13}}{13}$$"}
{"id":"604","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\triangle$$ABC, ∠ACB=90°, ∠B=30°, AC=2, D is a moving point on BC, EF is the perpendicular bisector of AD, intersecting AC at E and AB at F. Then the maximum value of BF is"},{"type":"image_path","image_path":"images/604_q0.png"}],"answer":"$$\\frac{8}{3}$$"}
{"id":"609","difficulty":"0.2","question_list":[{"type":"text","text":"In parallelogram ABCD, from point C, draw perpendiculars to AB and AD, with feet at F and E respectively. Connect BE and CF, intersecting at point G. Point N lies on CE, and connect FN intersecting BE at point M. Given ∠A = 120°, AE = 2ED, and CN = $$2 \\sqrt{3}$$, and FE = FN, then $$S_{\\triangle F M G}$$ ="},{"type":"image_path","image_path":"images/609_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{2}$$"}
{"id":"622","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the side length of square ABCD is 8, and P is a moving point on side CD. EF ⊥ BP intersects BP at G, and EF bisects the area of square ABCD. The minimum value of segment GC is   ."},{"type":"image_path","image_path":"images/622_q0.png"}],"answer":"$$2 \\sqrt{10} - 2 \\sqrt{2}$$/$$- 2 \\sqrt{2} + 2 \\sqrt{10}$$"}
{"id":"625","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$\\angle C = 60^\\circ$$, connect DB, $$D B \\bot B C$$, rotate $$\\triangle A D B$$ counterclockwise about point A to $$\\triangle A D^{'} B^{'}$$, draw $$B^{'} E \\parallel D B$$ intersecting line $$D^{'} D$$ at point E, connect $$B B^{'}$$ intersecting $$D^{'} E$$ at point F. If $$B^{'} E = 13 \\sqrt{3}$$, $$D F = 10$$, then $$A F =$$      ."},{"type":"image_path","image_path":"images/625_q0.png"}],"answer":"$$12 + 5 \\sqrt{3}$$/$$5 \\sqrt{3} + 12$$"}
{"id":"627","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCE, ∠B = ∠A, ∠E = 90°, point D lies on AB, and AD : BD = 5 : 11. Connect CD. If point D lies on the perpendicular bisector of CE and satisfies ∠A = 2∠BDC, and CE = 10, then the length of segment AB is      ."},{"type":"image_path","image_path":"images/627_q0.png"}],"answer":"20"}
{"id":"633","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠ABC = 45°, AD and BE are the altitudes to sides DC and AC respectively. Connect DE. From point D, draw DF ⊥ DE intersecting BE at point F. G is the midpoint of BE. Connect AF and DG. The relationship between AF and DG is    ."},{"type":"image_path","image_path":"images/633_q0.png"}],"answer":"$$A F = 2 D G$$,$$A F \\bot D G$$/$$A F \\bot D G$$,$$A F = 2 D G$$"}
{"id":"639","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 2, BC = 3, point E is the midpoint of side AB, connect CE, fold triangle BCE along CE to obtain triangle FCE, and let CF intersect BD at point P. Then the length of DP is    ."},{"type":"image_path","image_path":"images/639_q0.png"}],"answer":"$$\\frac{8 \\sqrt{13}}{17}$$"}
{"id":"640","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle$$ABC, AB=AC, ∠ABC=30°, D is a moving point on side AB (excluding point B). Construct a square CDEF with CD as one side, and connect BE. Let BD=x, and let the area of $$\\triangle$$BDE be s. If s=ax2+6x, where a is a constant, then the length of segment AB is     ."},{"type":"image_path","image_path":"images/640_q0.png"}],"answer":"$$\\left(\\right. \\frac{4}{3} a + \\frac{2}{3} \\left.\\right) x + 8$$."}
{"id":"644","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, diagonals AC and BD intersect at point O. F is a moving point on segment OD (point F does not coincide with points O or D). Connect CF, and from point F draw FG ⊥ CF intersecting AC and AB at points H and G, respectively. Connect CG intersecting BD at point M. Draw OE ∥ CD intersecting CG at point E, and let EF intersect AC at point N. The following conclusions are given: ① When BG = BM, AG = √2 BG; ② OH/OM = OF/OC; ③ When GM = HF, CF² = CN · BC; ④ CN² = BM² + DF². Among these, the correct ones are ______ (fill in the serial numbers only)."},{"type":"image_path","image_path":"images/644_q0.png"}],"answer":"①③④"}
{"id":"645","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rhombus ABCD, BC = 9, ∠ABC = 60°, point E lies on BC such that BE = 6. Triangle ABE is folded along AE to obtain triangle AB′E, where B′E intersects CD at point F. Then CF =            ."},{"type":"image_path","image_path":"images/645_q0.png"}],"answer":"$$\\frac{9}{5}$$"}
{"id":"648","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rhombus ABCD, AB = 2, DE ⊥ BC at point E, F is the midpoint of CD, connect AF and EF. If ∠AFE = 90°, then the length of CE is         ."},{"type":"image_path","image_path":"images/648_q0.png"}],"answer":"$$\\sqrt{3} - 1$$;"}
{"id":"649","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 5, BC = 8, point M is the midpoint of side BC, and P is a moving point on line AD. Construct right triangle MPQ on the right side of MP such that PM = PQ. Connect AM and AQ. The minimum perimeter of triangle AMQ is   ."},{"type":"image_path","image_path":"images/649_q0.png"}],"answer":"$$\\sqrt{221}$$+$$\\sqrt{41}$$"}
{"id":"650","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, E and F are moving points on ray $$BD$$, and $$\\angle EAF = 45^\\circ$$. Rays $$AE$$ and $$AF$$ intersect the extensions of $$BC$$ and $$CD$$ at G and H, respectively. Connect $$EC$$. Among the following conclusions: ① $$AE = CE$$; ② $$\\triangle AEF \\sim \\triangle GHC$$; ③ $$BG = GH + DH$$; ④ $$EF^2 = BE^2 + DF^2$$; ⑤ If $$AB = 3CH$$, then $$CD = 2CG$$; ⑥ $$S_{\\triangle AGH} : S_{\\triangle BCD} = GH : AB$$, the ones that are definitely correct are ______. (Write the correct sequence numbers on the line)"},{"type":"image_path","image_path":"images/650_q0.png"}],"answer":"①③④⑥"}
{"id":"657","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, E is a point on side BC. Connect AE. From point B, draw BF perpendicular to AE at point G, intersecting line CD at point F. Construct parallelogram BEHF with BE and BF as adjacent sides. Let M be the midpoint of BH. Connect GM. If AB = 3 and BC = 2, then the minimum value of GM is       ."},{"type":"image_path","image_path":"images/657_q0.png"}],"answer":"$$\\frac{2 \\sqrt{13}}{13}$$"}
{"id":"663","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, connect diagonals $$A C$$ and $$B D$$. Given $$A B = 6$$, $$A C = C D$$, $$\\angle A C B = 45 \\circ$$, $$\\angle A C D = 90 \\circ$$, the maximum value of diagonal $$B D$$ is      ."},{"type":"image_path","image_path":"images/663_q0.png"}],"answer":"$$6 \\sqrt{2} + 6$$"}
{"id":"665","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the acute vertex A of the isosceles right triangle ABC lies on a circle ⊙O, with ∠ACB = 90°. The legs AC and hypotenuse AB intersect ⊙O at points E and D, respectively. Tangents to ⊙O are drawn through points D and E, intersecting at point F, and point F happens to lie on leg BC. Connect OC, OD, and OE. If the radius of ⊙O is 4, then the maximum value of OC is        ."},{"type":"image_path","image_path":"images/665_q0.png"}],"answer":"$$2 \\sqrt{5} + 2$$"}
{"id":"671","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral ABCD is a square, E is any point on side BC, ∠AEF = 90°, and EF intersects the angle bisector CF of the exterior angle of the square at point F. From point C, draw CH ⊥ DF, intersecting the extension of DF at point H. If AB = 4 and BE = $$\\frac{1}{3}$$BC, then CH =     ."},{"type":"image_path","image_path":"images/671_q0.png"}],"answer":"$$\\frac{4 \\sqrt{5}}{5}$$"}
{"id":"672","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$ABC$$, D is the midpoint of $$AC$$, and P is a moving point on side $$AB$$. From point P, draw $$PE \\bot AB$$, intersecting $$BC$$ at point E. Connect $$DP$$ and $$DE$$. If $$AB = 8$$ and $$\\triangle PDE$$ is an isosceles triangle, then the length of $$BP$$ is                 ."},{"type":"image_path","image_path":"images/672_q0.png"}],"answer":"$$- 3 + \\sqrt{33}$$ or $$4$$ or $$12 - 4 \\sqrt{6}$$."}
{"id":"673","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD with side length 4, P is a moving point on side BC (excluding points B and C). Triangle ABP is folded along line AP, and point B lands at point E. On side CD, there is a point M such that when triangle CMP is folded along line MP, point C lands at point F on line PE. Line PE intersects CD at point N, and connect MA and NA. When triangle ABP is congruent to triangle ADN, the length of BP is ______."},{"type":"image_path","image_path":"images/673_q0.png"}],"answer":"$$4 \\sqrt{2} - 4$$"}
{"id":"674","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, point E lies on ray BC. Connect AB and DE, then the minimum value of $$\\frac{D E}{A E}$$ is     ."},{"type":"image_path","image_path":"images/674_q0.png"}],"answer":"$$\\frac{\\sqrt{5} - 1}{2}$$"}
{"id":"679","difficulty":"0.2","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, $$A B = 4 \\sqrt{2}$$, $$A C = 6$$, $$\\angle A = 45^\\circ$$. Fold $$\\triangle A B C$$ so that point $$C$$ lands on point $$D$$ on side $$A B$$, with crease $$E F$$ intersecting $$A C$$ at point $$E$$. As point $$D$$ moves continuously from $$B$$ toward $$A$$, the path length traversed by point $$E$$ is denoted as $$m$$. Then $$B C =$$     , $$\\text{m} =$$        ."},{"type":"image_path","image_path":"images/679_q0.png"}],"answer":"$$2 \\sqrt{5}$$     $$20 - 12 \\sqrt{2}$$"}
{"id":"696","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the area of rhombus $$A B C D$$ is $$8 \\sqrt{3}$$, $$\\angle B A D = 60 \\circ$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$. If point $$P$$ is a point on diagonal $$A C$$, then the minimum value of $$\\frac{1}{2} A P + B P$$ is       ."},{"type":"image_path","image_path":"images/696_q0.png"}],"answer":"$$\\text{2} \\sqrt{\\text{3}}$$"}
{"id":"699","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$\\Delta A B C$$, with circumcenter $$O$$, $$B C = 12$$, $$\\angle B A C = 60^\\circ$$, construct isosceles right triangles $$\\Delta A B D$$ and $$\\Delta A C E$$ outwardly on sides $$A B$$ and $$A C$$ respectively, connect $$B E$$ and $$C D$$ intersecting at point $$P$$, then the minimum value of $$O P$$ is      ."},{"type":"image_path","image_path":"images/699_q0.png"}],"answer":"$$6 - 2 \\sqrt{3}$$"}
{"id":"703","difficulty":"0.2","question_list":[{"type":"text","text":"Given: As shown in the figure, isosceles right triangle $$\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = A C$$, point D is a point outside $$\\triangle A B C$$, $$\\angle A D B = 45 \\circ$$, connect CD, $$A D = 4$$, $$C D = 5 \\sqrt{2}$$, the length of BC is        ."},{"type":"image_path","image_path":"images/703_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"708","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A D = 5, A B = 3$$, E is a moving point on $$B C$$, connect $$A E$$, construct $$D F \\bot A E$$ at F, connect $$C F$$, when $$\\triangle C D F$$ is an isosceles triangle, then the length of $$B E$$ is          ."},{"type":"image_path","image_path":"images/708_q0.png"}],"answer":"$$\\frac{5}{2}$$ or 4 or 1."}
{"id":"712","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C \\bot B C$$ at point $$C$$, $$\\angle B A C = \\angle A D C$$, and $$B C = \\frac{3}{4} A C$$. When $$C D = 4$$ and $$A D = 2$$, the length of segment $$B D$$ is      ."},{"type":"image_path","image_path":"images/712_q0.png"}],"answer":"$$\\frac{\\sqrt{109}}{2}$$"}
{"id":"724","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 9$$, point $$E$$ is a point on $$AD$$ such that $$AE : ED = 1 : 2$$, point $$P$$ is a moving point on side $$AB$$, connect $$PE$$, draw $$EF \\bot PE$$ intersecting ray $$BC$$ at point $$F$$, connect $$PF$$, point $$M$$ is the midpoint of $$PF$$, connect $$DM$$, then the minimum value of $$DM$$ is        ."},{"type":"image_path","image_path":"images/724_q0.png"}],"answer":"$$\\frac{21 \\sqrt{10}}{10}$$"}
{"id":"726","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, AB = AD = CD = 4, AD ∥ BC, ∠B = 60°, points E and F are two moving points on sides BC and CD respectively, and ∠EAF = 60°. Then the minimum area of △AEF is     ."},{"type":"image_path","image_path":"images/726_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"731","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 4$$, $$A C = 5$$, an equilateral triangle $$\\triangle B C D$$ is constructed outwardly on side $$B C$$, and $$A D$$ is connected. Then the maximum value of $$A D$$ is          ."},{"type":"image_path","image_path":"images/731_q0.png"}],"answer":"9"}
{"id":"732","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\Delta A B C$$, $$A B = 3$$, points $$D$$ and $$E$$ are moving points on sides $$B C$$ and $$C A$$ respectively, with $$B D = C E$$. Connect $$A D$$ and $$B E$$, intersecting at point $$F$$. When point $$D$$ moves from point $$B$$ to point $$C$$, the length of the path traced by point $$F$$ is         ."},{"type":"image_path","image_path":"images/732_q0.png"}],"answer":"$$\\frac{2 \\sqrt{3} \\pi}{3}$$"}
{"id":"736","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = BC = 5, AD ⊥ BC at point D, and AD = $$3$$. If point M is a moving point on AD, and point P starts from point B, moving along BM at a speed of $$1$$ unit per second, then along MA at a speed of $$\\sqrt{10}$$ units per second, the shortest time for point P to complete its motion is    seconds."},{"type":"image_path","image_path":"images/736_q0.png"}],"answer":"$$\\frac{3}{2} \\sqrt{10}$$"}
{"id":"748","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = 2, \\angle B A C = 120 \\circ$$, P is a moving point on side $$B C$$, connect $$A P$$, rotate segment $$A P$$ clockwise around point A by $$120 \\circ$$ to obtain $$A P^{'}$$, then the minimum value of segment $$P P^{'}$$ is       ."},{"type":"image_path","image_path":"images/748_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"752","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = AC, ∠ABC = α, tanα = $$\\frac{5}{6}$$, AD ⊥ BC at point D, point E is a moving point on segment AD, connect EB, rotate segment EB counterclockwise by 2α about point E to obtain segment EF, connect AF. If BC = 24, then the minimum value of segment AF is       ."},{"type":"image_path","image_path":"images/752_q0.png"}],"answer":"$$2 \\sqrt{61}$$"}
{"id":"764","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠C = 90°, AC = 8, BC = 16, point D is on side BC. Fold △ABC along DE such that point B coincides with point A. Connect AD. Point P lies on segment AD. When the distance from point P to the right-angle sides of △ABC is 5, the length of AP is     ."},{"type":"image_path","image_path":"images/764_q0.png"}],"answer":"$$\\frac{25}{3}$$ or $$\\frac{15}{4}$$."}
{"id":"766","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, O is the midpoint of BD. An equilateral triangle BCE is constructed inward on side BC. Connect AE and extend it to intersect CD at F. Connect BD, intersecting CE and AF at G and H, respectively. The following conclusions: ① $$\\angle C E H = 45 \\circ$$; ② $$B G = \\sqrt{2} D G$$; ③ $$G F // E D$$; ④ $$2 O H + D H = B D$$; ⑤ $$S_{\\triangle B E C} : S_{\\triangle B G C} = \\frac{\\sqrt{3} + 1}{2}$$, among which the correct ones are          ."},{"type":"image_path","image_path":"images/766_q0.png"}],"answer":"①③⑤"}
{"id":"770","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta A B C$$, $$A B = A C$$, $$B C = 12$$, $$D$$ is the midpoint of side $$A C$$, the perpendicular bisector of segment $$B D$$ intersects side $$B C$$ and $$A B$$ at points $$E$$ and $$F$$, respectively. Connect $$D F$$ and $$E F$$. Let $$B E = x$$, $$t a n \\angle A C B = y$$. The following conclusions are given: ① $$D F / / B C$$; ② The area of $$\\Delta B D E$$ is $$\\frac{3}{2} x y$$; ③ The perimeter of $$\\Delta C D E$$ is $$12 + x$$; ④ $$x^{2} - y^{2} = 9$$; ⑤ $$2 x - y^{2} = 9$$. Among these, the correct conclusions are       (fill in the serial numbers of all correct conclusions)."},{"type":"image_path","image_path":"images/770_q0.png"}],"answer":"②⑤"}
{"id":"777","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, AB = 12, points E, F, G, H lie on AB, BC, CD, DA respectively, EG = 13, FH = 15, then the area of quadrilateral EFGH is    ."},{"type":"image_path","image_path":"images/777_q0.png"}],"answer":"94.5"}
{"id":"778","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, given $$\\square A B C D$$, $$B E \\bot A D$$ at $$E$$, $$F$$ is the midpoint of $$A B$$, connect $$E F$$, translate $$\\Delta B E F$$ to the right to $$\\Delta H D G$$, such that $$B$$ coincides with $$H$$, $$E$$ coincides with $$D$$, and $$F$$ coincides with $$G$$, connect $$D F$$, $$H F$$, $$E H$$, if $$G$$ is the orthocenter of $$\\Delta D F H$$, $$A E = 2$$, $$B H = 8$$, then the distance from $$F$$ to $$E H$$ is        ."},{"type":"image_path","image_path":"images/778_q0.png"}],"answer":"3"}
{"id":"781","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, E is a point on CD. If triangle ADE is folded along line AE such that point D lands at point D' on side BC. F is a point on AD such that DF = CD'. EF intersects BD at point G, and AD' intersects BD at point H. D'E is parallel to BD, and HG = 4. Then BD =  ."},{"type":"image_path","image_path":"images/781_q0.png"}],"answer":"6+2$$\\sqrt{5}$$."}
{"id":"788","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4, A D = 6$$, $$E$$ and $$F$$ are the midpoints of $$A D$$ and $$B C$$ respectively, $$G$$ and $$H$$ lie on $$D C$$ and $$A B$$ respectively, and $$\\angle B E G = \\angle D F H = 90 \\circ$$. Connect $$B G$$ and $$D H$$. The perimeter of the hexagon $$I J K L M N$$, which is the overlapping region of $$\\triangle B E G$$ and $$\\triangle D F H$$, is"},{"type":"image_path","image_path":"images/788_q0.png"}],"answer":"9.8"}
{"id":"797","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, square paper ABCD is folded along line BE, and point C lands at point G. Connect BG and extend it to intersect CD at point H. Extend EG to intersect AD at point F, and connect FH. If AF = FD = 6 cm, then the length of FH is     cm."},{"type":"image_path","image_path":"images/797_q0.png"}],"answer":"3$$\\sqrt{5}$$"}
{"id":"801","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $ABCD$ with side length $4$, point $E$ is the quarter-point closer to point $D$, and point $F$ is the midpoint of $AB$. Fold $\\triangle AEF$ along $EF$ to obtain $\\triangle A'EF$, and connect $A'C$. Then the distance from point $D$ to the line containing $A'C$ is                ."},{"type":"image_path","image_path":"images/801_q0.png"}],"answer":"$$\\frac{28}{65} \\sqrt{65}$$"}
{"id":"804","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$\\triangle BPC$$ is an equilateral triangle, and the extensions of $$BP$$ and $$CP$$ intersect $$AD$$ at points $$E$$ and $$F$$, respectively. Connect $$BD$$ and $$DP$$, and let $$BD$$ intersect $$CF$$ at point $$H$$. The following conclusions are given:\n① $$\\triangle ABE \\cong \\triangle DCF$$; ② $$\\frac{FP}{PH} = \\frac{\\sqrt{3}}{3}$$; ③ $$DP^{2} = PH \\cdot PB$$; ④ $$\\frac{S_{\\triangle HPD}}{S_{\\text{square } ABCD}} = \\frac{\\sqrt{3} - 1}{2}$$, among which the correct ones are      . (Write all correct conclusion numbers)"},{"type":"image_path","image_path":"images/804_q0.png"}],"answer":"①②③."}
{"id":"810","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$C$$ is a moving point on segment $$A E$$ (not coinciding with points $$A$$ and $$E$$). On the same side of $$A E$$, construct equilateral triangle $$A B C$$ and equilateral triangle $$C D E$$. Let $$A D$$ intersect $$B E$$ at point $$O$$, $$A D$$ intersect $$B C$$ at point $$P$$, and $$B E$$ intersect $$C D$$ at point $$Q$$. Connect $$P Q$$. The following conclusions: ① $$A D = B E$$; ② $$P Q / / A E$$; ③ $$\\angle A O B = 60 \\circ$$; ④ $$\\triangle C P Q$$ is an equilateral triangle. The ones that always hold are      ."},{"type":"image_path","image_path":"images/810_q0.png"}],"answer":"①②③④"}
{"id":"811","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta A B C$$, $$A B = 10$$, $$A C = 8$$, $$B C = 6$$, a moving circle passing through point $$C$$ and tangent to side $$A B$$ intersects $$C A$$ and $$C B$$ at points $$P$$ and $$Q$$, respectively. The minimum length of segment $$P Q$$ is          ."},{"type":"image_path","image_path":"images/811_q0.png"}],"answer":"$$4.8$$"}
{"id":"812","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, the fixed line $$l$$ passes through the center $$O$$, and $$P$$ is a moving point on the radius $$OA$$. $$AC \\bot l$$ at point $$C$$. When the radius $$OA$$ rotates around point $$O$$, it always holds that $$OP = OC$$. When $$OA$$ rotates by $$60^\\circ$$ around point $$O$$, the ratio of the path lengths traveled by points $$P$$ and $$A$$ is ."},{"type":"image_path","image_path":"images/812_q0.png"}],"answer":"1."}
{"id":"821","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, given rectangle paper $$ABCD$$, $$AD = 15$$, $$AB = 8$$, point $$E$$ is on side $$BC$$. Fold the paper along $$AE$$ such that point $$B$$ lands at point $$F$$. Connect $$FC$$. When $$\\Delta EFC$$ is a right triangle, the area of $$\\Delta EFC$$ is       ."},{"type":"image_path","image_path":"images/821_q0.png"}],"answer":"28 or $$\\frac{108}{5}$$"}
{"id":"831","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠ABC = 90°, point D is the midpoint of AC. Draw the angle bisector DE of ∠ADB, intersecting AB at point E. AE = 6, DE = 10. Point P lies on side BC, and triangle DEP is an isosceles triangle. Find the length of BP:"},{"type":"image_path","image_path":"images/831_q0.png"}],"answer":"2 or 5 or 8 or 18"}
{"id":"832","difficulty":"0.2","question_list":[{"type":"text","text":"In square ABCD, point G is on AB, point H is on BC, and ∠GDH = 45°. DG and DH intersect diagonal AC at points E and F, respectively. The quantitative relationship among segments AE, EF, and FC is _______."},{"type":"image_path","image_path":"images/832_q0.png"}],"answer":"$$E F^{2} = A E^{2} + C F^{2}$$"}
{"id":"833","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, AB = 2. Extend BC to point E such that CE = 1, and connect DE. A moving point P starts from point A and moves along AB-BC-CD-DA toward point A at a speed of 1 unit per second. Let the motion time of point P be t seconds. When triangles ABP and DCE are congruent, the value of t is    ."},{"type":"image_path","image_path":"images/833_q0.png"}],"answer":"3 or 7"}
{"id":"842","difficulty":"0.2","question_list":[{"type":"text","text":"Given that the height of the equilateral triangle △ABC is 6, there is a point P in the plane of this triangle. If the distance from point P to line AB is 1, and the distance from point P to line AC is 3, then the distance from point P to line BC may be       ."},{"type":"image_path","image_path":"images/842_q0.png"}],"answer":"2, 4, 8, 10"}
{"id":"845","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, $$BD$$ is the diagonal of quadrilateral $$ABCD$$, $$BC = AD$$, $$\\angle A = \\angle CBD$$, $$\\angle ABD = 120^\\circ$$, $$AB = 3$$, $$CD = \\sqrt{19}$$, then the length of $$BC$$ is             ."},{"type":"image_path","image_path":"images/845_q0.png"}],"answer":"7"}
{"id":"847","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, ∠A = 60°, points E and F lie on sides AD and DC respectively, with DE = DF, and ∠EBF = 60°. If AE = 2 and FC = 3, then the length of EF is                 ."},{"type":"image_path","image_path":"images/847_q0.png"}],"answer":"$$\\sqrt{21}$$"}
{"id":"848","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, AB is the diameter of a semicircle, and C is a point on the semicircular arc. One side DG of square DEFG lies on the diameter AB, and the other side DE passes through the incenter O of triangle ABC, with point E lying on the semicircular arc.\n\n① If vertex F of the square also lies on the semicircular arc, then the ratio of the radius of the semicircle to the side length of the square is ______;\n\n② If the area of square DEFG is 100, and the inradius r of triangle ABC is 4, then the diameter AB of the semicircle is ______."},{"type":"image_path","image_path":"images/848_q0.png"}],"answer":"$$\\sqrt{5}$$:2     21"}
{"id":"849","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, points E and F are moving points on sides BC and CD of square ABCD with side length 2, and BE = CF. Connect DE and AF, intersecting at point P. From point P, draw PN perpendicular to CD at point N and PM perpendicular to BC at point M. Connect MN; find the minimum length of MN."},{"type":"image_path","image_path":"images/849_q0.png"}],"answer":"$$\\sqrt{5} - 1$$"}
{"id":"850","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, diagonals AC and BD intersect at point E. If AC bisects ∠DAB, and AB = AD, CD = CB, the following four conclusions hold: ① AC ⊥ BD; ② BE = DE; ③ ∠DAB = 2∠BAC; ④ △ABD is an equilateral triangle. Please write the serial numbers of the correct conclusions"},{"type":"image_path","image_path":"images/850_q0.png"}],"answer":"①②③"}
{"id":"851","difficulty":"0.2","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, AC = BC, ∠ACB = 90°, point D is a point on side AB. CD is rotated 90° clockwise about point D to DE, and CE intersects AB at point G. Given that AD = 8, BG = 6, point F is the midpoint of AE. Connect DF. Find the length of segment DF."},{"type":"image_path","image_path":"images/851_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"875","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$D E$$ bisects $$\\angle A D C$$, $$A D = 6$$, $$B E = 2$$, then the perimeter of parallelogram $$A B C D$$ is      ."},{"type":"image_path","image_path":"images/875_q0.png"}],"answer":"20"}
{"id":"896","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, points B, F, C, E lie on a straight line, $$BF = CE$$, $$\\angle B = \\angle E$$, add one condition to make $$\\triangle ABC \\cong \\triangle DEF$$, this added condition can be      (only one is required, no auxiliary lines needed)."},{"type":"image_path","image_path":"images/896_q0.png"}],"answer":"$$A B = D E$$ or $$∠A=∠D$$ or $$∠ACB=∠EFD$$ or $$AC∥DF$$, the answer is not unique."}
{"id":"913","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the median of $$\\triangle ABC$$, $$BE$$ is the median of $$\\triangle ABD$$, $$EF \\bot BC$$ at point $$F$$. If $$S_{\\triangle ACD} = 12 \\text{cm}^{2}$$, $$BD = 4 \\text{cm}$$, then the length of $$EF$$ is        $$\\text{cm}$$."},{"type":"image_path","image_path":"images/913_q0.png"}],"answer":"3"}
{"id":"927","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$MN$$ is a chord of $$\\bigodot O$$, $$\\angle N = 50^\\circ$$, then the measure of $$\\angle MON$$ is      ."},{"type":"image_path","image_path":"images/927_q0.png"}],"answer":"$$80 \\circ$$"}
{"id":"946","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of ⊙$$O$$, $$AC$$ is the tangent to ⊙$$O$$, with $$A$$ as the point of tangency. Connect $$BC$$, which intersects ⊙$$O$$ at point $$D$$, and connect $$OD$$. If $$\\angle AOD = 82^\\circ$$, then $$\\angle C =$$         $$^\\circ$$."},{"type":"image_path","image_path":"images/946_q0.png"}],"answer":"49"}
{"id":"975","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, the perpendicular bisector of $$A C$$ intersects $$B C$$ at point D and $$A C$$ at point E. Connect $$A D$$. If the perimeter of $$\\triangle A B C$$ is 13 and $$A E = 2$$, then the perimeter of $$\\triangle A B D$$ is    ."},{"type":"image_path","image_path":"images/975_q0.png"}],"answer":"9"}
{"id":"996","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 10$$, $$B C = 14$$, points D and E are the midpoints of sides $$A B$$ and $$A C$$ respectively, point F is a point on segment $$D E$$, connect $$A F$$ and $$B F$$, if $$\\angle A F B = 90^\\circ$$, then the length of segment $$E F$$ is        ."},{"type":"image_path","image_path":"images/996_q0.png"}],"answer":"2"}
{"id":"998","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = 4$$, $$B C = 6$$, the perpendicular bisector of $$A C$$ intersects $$A D$$ at point $$E$$, and connect $$C E$$. The perimeter of $$\\triangle C D E$$ is       ."},{"type":"image_path","image_path":"images/998_q0.png"}],"answer":"10"}
{"id":"1020","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the side lengths of $$\\triangle A B C$$ are $$A B$$, $$B C$$, $$C A$$, which are 10, 15, 20 respectively. Its three angle bisectors intersect at point O. Connect OA, OB, OC, dividing $$\\triangle A B C$$ into three triangles. Then $$S_{\\triangle A B O} : S_{\\triangle B C O} : S_{\\triangle C A O}$$ equals          ."},{"type":"image_path","image_path":"images/1020_q0.png"}],"answer":"2:3:4"}
{"id":"1071","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A = 90 \\circ$$, point D lies on side AB, and CD is connected. If $$B D = C D$$ and $$\\frac{A D}{B D} = \\frac{1}{3}$$, then $$t a n B =$$     ."},{"type":"image_path","image_path":"images/1071_q0.png"}],"answer":"$$\\frac{\\sqrt{2}}{2}$$/$$\\frac{1}{2} \\sqrt{2}$$"}
{"id":"1075","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AC = 2, BC = 4, point O lies on BC, and a circle with radius OB is tangent to AC at point A. D is a moving point on side BC. When △ACD is a right triangle, the length of AD is           ."},{"type":"image_path","image_path":"images/1075_q0.png"}],"answer":"$$\\frac{3}{2}$$ or $$\\frac{6}{5}$$"}
{"id":"1084","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the radius of $$\\bigodot O$$ is $$2$$ $$\\text{cm}$$, $$AB$$ is a chord of $$\\bigodot O$$, and point $$C$$ is a point on $$\\overset{⌢}{AB}$$. When $$\\overset{⌢}{AB}$$ is folded along chord $$AB$$, point $$C$$ coincides with the center $$O$$. The area of the shaded region is       . (Express the result in terms of $$\\pi$$ and radicals)"},{"type":"image_path","image_path":"images/1084_q0.png"}],"answer":"$$\\left(\\frac{2}{3} \\pi - \\sqrt{3}\\right) \\text{cm}^{2}$$"}
{"id":"1094","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, rectangle paper $$ABCD$$ is folded along $$EF$$, and after folding, point $$A$$ lands at $$A^{'}$$, and point $$B$$ coincides exactly with point $$D$$. Given that $$\\angle DFC = 60^\\circ$$, $$CF = 3$$, the length of $$AE$$ is      ."},{"type":"image_path","image_path":"images/1094_q0.png"}],"answer":"$$3$$"}
{"id":"1131","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A D$$ is the angle bisector of $$\\triangle A B C$$, $$D E \\bot A B$$ at point E, $$C D = 3$$, $$A B = 10$$, then the area of $$\\triangle A B D$$ is          ."},{"type":"image_path","image_path":"images/1131_q0.png"}],"answer":"15"}
{"id":"1139","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the left and right sides of the alley are vertical walls. When the ladder leans against the left wall, the distance from the bottom of the ladder to the left wall corner is $$0.7$$ meters, and the top of the ladder is 2.4 meters above the ground. If the position of the bottom of the ladder remains unchanged and the ladder leans against the right wall, the top of the ladder is 2 meters above the ground. The width of the alley is      meters."},{"type":"image_path","image_path":"images/1139_q0.png"}],"answer":"$$2.2$$/$$2 \\frac{1}{5}$$/$$\\frac{11}{5}$$"}
{"id":"1146","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\text{Rt} \\triangle A B C$$, squares are constructed outwardly on each of its three sides, with areas denoted as $$S_{1}, S_{2}, S_{3}$$ respectively. If $$S_{3} + S_{2} - S_{1} = 18$$, then the area of the shaded region in the figure is"},{"type":"image_path","image_path":"images/1146_q0.png"}],"answer":"$$\\frac{9}{2}$$"}
{"id":"1179","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 8 \\text{cm}$$, $$B C = 6 \\text{cm}$$. With point $$B$$ as the center, draw an arc with any length as the radius, intersecting sides $$A B$$ and $$B C$$ at points $$D$$ and $$E$$, respectively; then, with $$D$$ and $$E$$ as centers, draw arcs with radii greater than $$\\frac{1}{2} D E$$, intersecting at point $$F$$; draw ray $$B F$$ intersecting side $$A C$$ at point $$G$$. If the area of $$\\triangle A B G$$ is $$20 \\text{cm}^{2}$$, then the area of $$\\triangle C B G$$ is      $$\\text{cm}^{2}$$."},{"type":"image_path","image_path":"images/1179_q0.png"}],"answer":"15"}
{"id":"1211","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point D is a moving point on $$A B$$, connect $$C D$$, point E is a point on segment $$C D$$, points F and G are moving points on sides $$C A$$ and $$C B$$ respectively. If $$\\angle A C B = 30 \\circ$$, $$C E = 6$$, then the minimum perimeter of $$\\triangle E F G$$ is     ."},{"type":"image_path","image_path":"images/1211_q0.png"}],"answer":"6"}
{"id":"1223","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, arrange a set of triangle boards as shown in the figure, in which $$\\angle 1 =$$        ."},{"type":"image_path","image_path":"images/1223_q0.png"}],"answer":"$$120^{\\circ}$$"}
{"id":"1228","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$B E$$ and $$C E$$ are the angle bisectors of $$\\angle A B C$$ and $$\\angle A C B$$, respectively, $$E D \\parallel A C$$ intersects $$B C$$ at point D, and $$E F \\bot A B$$ at point F. If $$B C = 35$$, $$E F = 5$$, $$D E = 13$$, then the area of $$\\triangle E B D$$ is        ."},{"type":"image_path","image_path":"images/1228_q0.png"}],"answer":"$$55$$"}
{"id":"1253","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, which is a pattern formed by two set squares, the measure of $$\\angle A E D$$ is      $$\\circ$$."},{"type":"image_path","image_path":"images/1253_q0.png"}],"answer":"$$105$$"}
{"id":"1256","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$C A = C B$$, $$A D$$ bisects $$\\angle C A B$$ and intersects $$B C$$ at D, $$D E \\bot A B$$ at E. If $$A B = 6$$, the perimeter of $$\\triangle D E B$$ is       ."},{"type":"image_path","image_path":"images/1256_q0.png"}],"answer":"6"}
{"id":"1280","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, points $$B$$, $$E$$, $$C$$, $$F$$ lie on a straight line, $$AC \\parallel DF$$, $$BE = CF$$. Only one additional condition is needed to prove $$\\triangle ABC \\approx \\triangle DEF$$; this condition can be        (write one such condition)."},{"type":"image_path","image_path":"images/1280_q0.png"}],"answer":"$$A C = D F$$ or $$\\angle A = \\angle D$$ or $$\\angle A B C = \\angle D E F$$ or $$A B \\parallel D E$$ (The answer is not unique)."}
{"id":"1292","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, construct squares $$A C D E$$, $$C B G F$$, and $$A H I B$$ outwardly on the three sides of $$\\triangle A B C$$. Let P be a point on $$H I$$. Denote the areas of squares $$A C D E$$ and $$A H I B$$ as $$S_{1}$$ and $$S_{2}$$, respectively. If $$S_{1} = 16$$ and $$S_{2} = 25$$, then the area of quadrilateral $$A C B P$$ is          ."},{"type":"image_path","image_path":"images/1292_q0.png"}],"answer":"18.5"}
{"id":"1298","difficulty":"0.8","question_list":[{"type":"text","text":"Place a set of triangle rulers as shown in the figure; then $$\\angle A O B$$ equals   ."},{"type":"image_path","image_path":"images/1298_q0.png"}],"answer":"$$105 \\circ$$"}
{"id":"1300","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C, B D$$ intersect at point $$O$$, $$A C = 16$$, $$B D = 12$$, point $$E$$ is the midpoint of $$C D$$, connect $$O E$$, then the length of $$O E$$ is        ."},{"type":"image_path","image_path":"images/1300_q0.png"}],"answer":"5"}
{"id":"1310","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, with endpoints $$A$$ and $$B$$ of segment $$AB$$ as centers, arcs are drawn with radius greater than $$\\frac{1}{2} AB$$, intersecting at points $$M$$ and $$N$$. Draw line $$MN$$, and let point $$C$$ be a point on line $$MN$$. Connect $$CB$$ and $$CA$$. With $$C$$ as center and $$CB$$ as radius, draw an arc intersecting the extension of $$AC$$ at point $$D$$. Connect $$BD$$. If $$\\angle BDC = 25^\\circ$$, then the measure of $$\\angle BAC$$ is             ."},{"type":"image_path","image_path":"images/1310_q0.png"}],"answer":"65°"}
{"id":"1326","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$A C = 8$$, $$A E \\bot B D$$, with foot at $$E$$, if $$\\angle A B D = 2 \\angle C B D$$, then the length of $$B E$$ is       ."},{"type":"image_path","image_path":"images/1326_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"1330","difficulty":"0.8","question_list":[{"type":"text","text":"Given: As shown in the figure, in $$\\triangle A B C$$, $$B O, C O$$ are the angle bisectors of $$\\angle A B C$$ and $$\\angle A C B$$, respectively. A line passing through point O intersects $$A B$$ and $$A C$$ at points D and E, respectively, and $$D E / / B C$$. If $$A B = 6 \\text{cm}, A C = 8 \\text{cm}$$, then the perimeter of $$\\triangle A D E$$ is      ."},{"type":"image_path","image_path":"images/1330_q0.png"}],"answer":"$$14 \\text{cm}$$"}
{"id":"1357","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$B A = B D$$. With point $$A$$ as the center and $$A D$$ as the radius, draw an arc intersecting $$B D$$ at another point $$F$$. Then, with points $$D$$ and $$F$$ as centers, draw arcs with radii greater than $$\\frac{1}{2} D F$$, intersecting at point $$G$$. Draw ray $$A G$$ intersecting $$B D$$ at point $$E$$. If $$\\angle C = 70^\\circ$$, then the measure of $$\\angle D A E$$ is $$\\circ$$."},{"type":"image_path","image_path":"images/1357_q0.png"}],"answer":"20"}
{"id":"1367","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, points D, E, F are the midpoints of sides $$B C$$, $$A D$$, and $$C E$$ respectively, and $$S_{\\triangle A B C} = 16$$. Then the area of the shaded region $$S_{shade} =$$       ."},{"type":"image_path","image_path":"images/1367_q0.png"}],"answer":"4"}
{"id":"1412","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, if $$A C = A D = D B$$, and $$\\angle B = 24^\\circ$$, then $$\\angle B A C =$$        ."},{"type":"image_path","image_path":"images/1412_q0.png"}],"answer":"$$108 \\circ$$"}
{"id":"1467","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$C D$$ is the altitude, if $$\\angle B = 58 \\circ$$, then the measure of $$\\angle A C D$$ is      ."},{"type":"image_path","image_path":"images/1467_q0.png"}],"answer":"$$58 \\circ$$"}
{"id":"1470","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, FE ∥ ON, OE bisects ∠MON, ∠FEO = 28°, then ∠MFE =  degrees."},{"type":"image_path","image_path":"images/1470_q0.png"}],"answer":"56"}
{"id":"1478","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the regular pentagon $$A B C D E$$, extend $$A E$$ and $$C D$$ to intersect at point $$F$$. Then the measure of $$\\angle F$$ is      $$\\circ$$."},{"type":"image_path","image_path":"images/1478_q0.png"}],"answer":"$$36$$"}
{"id":"1518","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, an equilateral triangle ABC with side length $$5 \\text{cm}$$ is translated 1 cm to the right to obtain the equilateral triangle A'B'C'. At this time, the perimeter of the shaded region is      $$\\text{cm}$$."},{"type":"image_path","image_path":"images/1518_q0.png"}],"answer":"$$12$$"}
{"id":"1524","difficulty":"0.8","question_list":[{"type":"text","text":"In the isosceles right triangle $$\\triangle A B C$$, $$A C = B C = \\sqrt{3}, \\angle A C B = 90^{\\circ}$$, D is a point on $$A B$$ such that $$A D = 2$$; an arc is drawn with center $$A$$ and radius $$A D$$, intersecting $$B C$$ at point $$F$$ and the extension of $$A C$$ at point $$E$$. The area of the shaded region in the figure is               ."},{"type":"image_path","image_path":"images/1524_q0.png"}],"answer":"$$\\frac{2 \\pi - 2 \\sqrt{3}}{6}$$"}
{"id":"1536","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the square $$ABCD$$ has side length 4, and point $$E$$ is a point on side $$AB$$ (point $$E$$ coincides with point $$A$$ or $$B$$). From point $$A$$, draw $$AF \\bot DE$$, with foot at $$G$$, and $$AF$$ intersects side $$BC$$ at point $$F$$. Connect $$DF$$ and $$EF$$. If the area of $$\\triangle DEF$$ is $$\\frac{13}{2}$$, then the length of $$AE$$ is        ."},{"type":"image_path","image_path":"images/1536_q0.png"}],"answer":"3 or 1"}
{"id":"1545","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, a sector with a central angle of $$90 \\circ$$ is cut out from a circular patch with a radius of 2. The area of this sector is        ."},{"type":"image_path","image_path":"images/1545_q0.png"}],"answer":"$$2 \\pi$$"}
{"id":"1546","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = 3$$, $$B C = 4$$, an arc is drawn with point A as the center and $$A C$$ as the radius, intersecting $$A B$$ at point D, with $$B D = 2$$. Then $$\\angle A C B =$$        °."},{"type":"image_path","image_path":"images/1546_q0.png"}],"answer":"90"}
{"id":"1575","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a rectangle, and diagonals $$A C$$, $$B D$$ intersect at point $$O$$. If $$\\angle A O B = 50^{\\circ}$$, then $$\\angle O C D =$$           ."},{"type":"image_path","image_path":"images/1575_q0.png"}],"answer":"$$65 \\circ$$"}
{"id":"1656","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ and $$AC$$ are tangents to $$\\bigodot O$$, with B and C as the points of tangency. Connect $$OC$$ and extend it to D such that $$CD = OC$$. Connect $$AD$$. If $$\\angle BAD = 75^\\circ$$, then the measure of $$\\angle AOC$$ is      ."},{"type":"image_path","image_path":"images/1656_q0.png"}],"answer":"$$65 \\circ$$"}
{"id":"1682","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in right triangle $$\\text{R} \\triangle A B C$$, $$\\angle C = 90^\\circ$$, $$\\angle B = 60^\\circ$$, point $$D$$ lies on $$B C$$, $$B D = 4$$, points $$P$$ and $$E$$ are moving points on $$A C$$ and $$A B$$ respectively. When the value of $$D P + E P$$ is minimized, $$B E = 5$$. Find the length of $$A B$$."},{"type":"image_path","image_path":"images/1682_q0.png"}],"answer":"$$14$$"}
{"id":"1717","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$M$$ and $$N$$ are the midpoints of sides $$AC$$ and $$BC$$ of $$\\triangle ABC$$, respectively, and $$AN$$ intersects $$BM$$ at point $$O$$. Then $$\\frac{\\text{area of } \\triangle BON}{\\text{area of } \\triangle ABC} =$$      ."},{"type":"image_path","image_path":"images/1717_q0.png"}],"answer":"$$\\frac{1}{6}$$"}
{"id":"1723","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle A = 50 \\circ$$, points D, E, F lie on sides $$B C$$, $$A B$$, $$A C$$ respectively. If $$B D = C F$$, $$B E = C D$$, then $$\\angle E D F =$$      degrees."},{"type":"image_path","image_path":"images/1723_q0.png"}],"answer":"$$65 \\circ$$"}
{"id":"1731","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$ and $$\\triangle D E F$$, points B, F, C, E lie on the same straight line, $$A B = D E$$, $$\\angle B = \\angle E$$. Add one condition such that $$\\triangle A B C ≌ \\triangle D E F$$. The added condition can be      (only one is required, no auxiliary lines needed);"},{"type":"image_path","image_path":"images/1731_q0.png"}],"answer":"$$\\angle A = \\angle D$$ (The answer is not unique)"}
{"id":"1737","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the angle bisector of $$\\triangle ABC$$, $$DE \\bot AB$$ at point $$E$$, if $$AC = 6$$, $$DE = 2$$, then the area of $$\\triangle ACD$$ is      ."},{"type":"image_path","image_path":"images/1737_q0.png"}],"answer":"6"}
{"id":"1746","difficulty":"0.8","question_list":[{"type":"text","text":"In equilateral triangle △ABC, P is a point on BC, and D is a point on AC, such that ∠APD = 60°, BP = 4, CD = 2. Then the side length of △ABC is    ."},{"type":"image_path","image_path":"images/1746_q0.png"}],"answer":"8"}
{"id":"1751","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, equilateral triangle $$A B C$$, $$P$$ is a point on $$B C$$, and $$\\angle 1 = \\angle 2$$, then the measure of $$\\angle 3$$ is          (degrees)."},{"type":"image_path","image_path":"images/1751_q0.png"}],"answer":"$$60$$"}
{"id":"1754","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the equilateral triangle $$\\triangle A B C$$ has side length 3, point $$P$$ is a point on side $$B C$$ such that $$B P = 1$$, and point $$D$$ is a point on side $$A C$$. If $$\\angle A P D = 60^\\circ$$, then the length of $$C D$$ is        ."},{"type":"image_path","image_path":"images/1754_q0.png"}],"answer":"$$\\frac{\\text{2}}{\\text{3}}$$"}
{"id":"1764","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 8$$, $$B E \\bot A C$$, then $$B E =$$      ."},{"type":"image_path","image_path":"images/1764_q0.png"}],"answer":"4.8"}
{"id":"1781","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = BC, AC = 2 cm, $$S_{\\triangle A B C} = 3 \\text{cm}^{2}$$, the perpendicular bisector of side BC is l, point D is the midpoint of side AC, and point P is a moving point on l. Then the minimum perimeter of △PCD is      ."},{"type":"image_path","image_path":"images/1781_q0.png"}],"answer":"4"}
{"id":"1786","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$BO$$ bisects $$\\angle CBA$$, $$CO$$ bisects $$\\angle ACB$$, and $$MN \\parallel BC$$. Let $$AB = 12$$, $$BC = 24$$, $$AC = 18$$; then the perimeter of $$\\triangle AMN$$ is       ."},{"type":"image_path","image_path":"images/1786_q0.png"}],"answer":"30"}
{"id":"1794","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle A = 36 \\circ$$, $$B D$$ bisects $$\\angle A B C$$ and intersects $$A C$$ at point D, point E is the midpoint of $$A B$$, and connect $$D E$$. Then the measure of $$\\angle A D B$$ is      ."},{"type":"image_path","image_path":"images/1794_q0.png"}],"answer":"$$108 \\circ$$"}
{"id":"1841","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, point E lies on the side AD of rectangle $$A B C D$$, and $$B C = E C = 8$$, $$\\angle A B E = 15^\\circ$$. Then the length of $$A B$$ is          ."},{"type":"image_path","image_path":"images/1841_q0.png"}],"answer":"4"}
{"id":"1851","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C$$, $$B D$$ intersect at point O, $$B D = 8$$, $$O E \\parallel A B$$, intersecting $$B C$$ at point E, $$O E = 2.5$$, then the length of $$A C$$ is       ."},{"type":"image_path","image_path":"images/1851_q0.png"}],"answer":"6"}
{"id":"1905","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an inscribed triangle of $$\\bigodot O$$. If $$\\angle O B C = 28^\\circ$$, then $$\\angle A =$$      ."},{"type":"image_path","image_path":"images/1905_q0.png"}],"answer":"$$62 \\circ$$"}
{"id":"1910","difficulty":"0.8","question_list":[{"type":"text","text":"In his work \"Ping Sanchao Ju Yao\", the early Qing mathematician Mei Wending provided a complete proof of Qin Jiushao's \"Sanxie Qiuji Shu\", a method for calculating the area of a triangle proposed by the Southern Song mathematician Qin Jiushao. In the proof, he creatively designed a right triangle and derived the conclusion: as shown in the figure, $$AD$$ is the altitude of the acute triangle $$\\triangle ABC$$, then $$BD = \\frac{1}{2} \\left(BC + \\frac{AB^{2} - AC^{2}}{BC}\\right)$$. When $$AB = 7$$, $$BC = 6$$, and $$AC = 5$$, $$CD =$$    ."},{"type":"image_path","image_path":"images/1910_q0.png"}],"answer":"$$1$$"}
{"id":"1912","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a square, $$\\triangle A D E$$ is an equilateral triangle, $$E F \\bot A B$$ at point F. If $$A D = 4$$, then $$E F =$$      ."},{"type":"image_path","image_path":"images/1912_q0.png"}],"answer":"2"}
{"id":"1921","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$O$$ is the midpoint of $$B D$$, and line $$E F$$ passes through point $$O$$ and intersects $$A B$$ and $$C D$$ at points $$E$$ and $$F$$, respectively. If $$A E = 10$$, then the length of $$C F$$ is           ."},{"type":"image_path","image_path":"images/1921_q0.png"}],"answer":"10"}
{"id":"1924","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠ACB = 90°, $$AC = BC = 2 \\sqrt{2}$$, point D is the midpoint of AB, and point P lies on AC such that CP = 1. Rotate CP around point C in the plane; the corresponding point of P is point Q. Connect AQ and DQ. When ∠ADQ = 90°, the length of AQ is      ."},{"type":"image_path","image_path":"images/1924_q0.png"}],"answer":"$$\\sqrt{5}$$ or $$\\sqrt{13}$$/$$\\sqrt{13}$$ or $$\\sqrt{5}$$"}
{"id":"1930","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, D is a point on $$A B$$, $$C F \\parallel A B$$, and points D, E, F are collinear. Please add one condition   , such that $$A E = C E$$. (Only one case needs to be added.)"},{"type":"image_path","image_path":"images/1930_q0.png"}],"answer":"$$D E = E F$$ or $$A D = C F$$ (The answer is not unique.)"}
{"id":"1947","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, if $$D E \\parallel B C, F G \\parallel A C, \\angle B D E = 120 \\circ, \\angle D F G = 115 \\circ$$, then $$\\angle C =$$        °."},{"type":"image_path","image_path":"images/1947_q0.png"}],"answer":"$$5 5 \\circ$$"}
{"id":"1956","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, a rectangular paper $$ABCD$$ is folded along crease $$EF$$. After folding, the image of side $$EC$$, denoted as $$EH$$, passes through point $$A$$, and the image of side $$CD$$, denoted as $$HG$$, intersects the extension of $$BA$$ at point $$P$$. If $$PA = PG$$, $$AH = BE$$, and $$CD = 3$$, then the length of $$BC$$ is         ."},{"type":"image_path","image_path":"images/1956_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"2003","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle D A B = 40 \\circ$$, connect $$A C$$, and with point $$A$$ as the center and $$A C$$ as the radius, draw an arc intersecting line $$A D$$ at point $$E$$, connect $$C E$$, then the measure of $$\\angle A E C$$ is        ."},{"type":"image_path","image_path":"images/2003_q0.png"}],"answer":"$$10 \\circ$$ or $$80 \\circ$$"}
{"id":"2016","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, $$A B = 2$$, $$B D$$ is the altitude from $$B$$ to side $$A C$$, and $$B C$$ is extended to point $$E$$ such that $$C E = C D$$. Then the length of $$B E$$ is            ."},{"type":"image_path","image_path":"images/2016_q0.png"}],"answer":"3"}
{"id":"2089","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the altitude of the equilateral $$\\triangle ABC$$, and M, N are moving points on segments $$AD$$ and $$AC$$ respectively, with $$AM = BN$$. When $$BM + CN$$ is minimized, $$\\angle ANC =$$            ."},{"type":"image_path","image_path":"images/2089_q0.png"}],"answer":"$$105 \\circ$$"}
{"id":"2092","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, if $$B A = B D$$, $$A E \\bot B D$$, $$\\angle C = 70^\\circ$$, then $$\\angle D A E =$$         degrees."},{"type":"image_path","image_path":"images/2092_q0.png"}],"answer":"$$20$$"}
{"id":"2118","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$BD$$ is the median of $$\\triangle ABC$$, $$AB = 6$$, $$BC = 4$$, the difference in the perimeters of $$\\triangle ABD$$ and $$\\triangle BCD$$ is      ."},{"type":"image_path","image_path":"images/2118_q0.png"}],"answer":"2"}
{"id":"2156","difficulty":"0.8","question_list":[{"type":"text","text":"In rectangle $$A B C D$$, $$A B = 3$$, $$A D = 2 \\sqrt{13}$$, point M is on side $$B C$$, connect $$A M$$, fold $$\\triangle A B M$$ along $$A M$$ to obtain $$\\triangle A B^{'} M$$, $$B^{'} M$$ intersects $$A D$$ at point N, if point N is the midpoint of $$A D$$, then the length of $$B M$$ is                   ."},{"type":"image_path","image_path":"images/2156_q0.png"}],"answer":"$$2 + \\sqrt{13}$$/$$\\sqrt{13} + 2$$"}
{"id":"2163","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, and point D is any point on side $$B C$$ other than B and C. $$D E \\bot A B$$ at point E, and $$D F \\bot A C$$ at point F. If the altitude from side $$B C$$, $$A M = 10$$, then $$D E + D F =$$           ."},{"type":"image_path","image_path":"images/2163_q0.png"}],"answer":"10"}
{"id":"2179","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, △ABC ≌ △DBE, the perimeter of △ABC is 30, AB = 9, BE = 8, then the length of AC is  ."},{"type":"image_path","image_path":"images/2179_q0.png"}],"answer":"13"}
{"id":"2211","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the median of $$\\triangle ABC$$. If $$S_{\\triangle ABC} = 2$$, then $$S_{\\triangle ACD} =$$        ."},{"type":"image_path","image_path":"images/2211_q0.png"}],"answer":"$$1$$"}
{"id":"2223","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle \\text{ABC}$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A B C = 60 \\circ$$, $$B C = 2$$. If a semicircle is drawn with $$A B$$ as the diameter, and an arc is drawn with point $$B$$ as the center and $$B C$$ as the radius, intersecting $$A B$$ at point $$D$$, then the area of the shaded region is                     . (Express the result in terms of $$\\pi$$)"},{"type":"image_path","image_path":"images/2223_q0.png"}],"answer":"$$\\frac{2}{3}$$π+$$\\sqrt{3}$$"}
{"id":"2233","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$AF \\parallel CD$$, $$BD$$ bisects $$\\angle EBF$$, and $$BC \\bot BD$$. The following conclusions: ① $$BC$$ bisects $$\\angle ABE$$; ② $$AC \\parallel BE$$; ③ $$\\angle CBE + \\angle D = 90^\\circ$$; ④ $$\\angle DEB = 2 \\angle BCD$$. Among these, the correct conclusions are        (only fill in the serial numbers)."},{"type":"image_path","image_path":"images/2233_q0.png"}],"answer":"①③④"}
{"id":"2236","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\bigodot O$$, $$A$$ is a point on the major arc $$BC$$, $$\\angle BAC = \\alpha$$, connect $$BO$$ and $$CO$$, extend $$BO$$ to intersect $$AC$$ at point $$D$$, then the angle in the figure that measures $$2\\alpha$$ is      ."},{"type":"image_path","image_path":"images/2236_q0.png"}],"answer":"$$\\angle B O C$$"}
{"id":"2244","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, a triangular cardboard with an area of 10, points D, E, F are the midpoints of segments AF, BD, CE respectively, then the area of the shaded region is      ."},{"type":"image_path","image_path":"images/2244_q0.png"}],"answer":"$$\\frac{10}{7}$$/$$1 \\frac{3}{7}$$"}
{"id":"2264","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, if $$\\triangle A B C \\approx \\triangle E B D$$, and $$B D = 4, A B = 8$$, then the area of the shaded region $$S_{\\triangle A C E} =$$        ."},{"type":"image_path","image_path":"images/2264_q0.png"}],"answer":"16"}
{"id":"2281","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$Rt \\triangle ABC$$, $$\\angle ACB = 90^\\circ$$, $$D$$ is the midpoint of $$AB$$, $$CD = 5$$, then $$AB =$$      ."},{"type":"image_path","image_path":"images/2281_q0.png"}],"answer":"10"}
{"id":"2283","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $$AC, BD$$ of rhombus $$ABCD$$ intersect at point $$O$$, $$AE \\bot BC$$, with foot at $$E$$. If $$AC = 6, BD = 8$$, then the length of $$AE$$ is      ."},{"type":"image_path","image_path":"images/2283_q0.png"}],"answer":"$$\\frac{24}{5}$$"}
{"id":"2284","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C, B D$$ intersect at point O, $$\\angle A O D = 116 \\circ$$, the measure of $$\\angle A C D$$ is         ."},{"type":"image_path","image_path":"images/2284_q0.png"}],"answer":"$$58 \\circ$$"}
{"id":"2331","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in right triangles Rt△BAC and Rt△BDC, ∠BAC = ∠BDC = 90°, O is the midpoint of BC, and AO and DO are connected. If AO = 3, then the length of DO is     ."},{"type":"image_path","image_path":"images/2331_q0.png"}],"answer":"3"}
{"id":"2341","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, diagonals $$A C$$, $$B D$$ intersect at point O, and $$A E$$ perpendicularly bisects $$O B$$ at point E. Then the length of $$A D$$ is            ."},{"type":"image_path","image_path":"images/2341_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"2348","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$D$$, $$E$$, $$F$$ are the midpoints of $$A B$$, $$B C$$, $$C A$$ respectively. If $$D E = 3$$, then $$B F =$$        ."},{"type":"image_path","image_path":"images/2348_q0.png"}],"answer":"3"}
{"id":"2405","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$\\triangle ABC$$, $$AB = AC = 8$$, $$\\angle BAC = 90^\\circ$$, the circle $$\\bigodot O$$ with diameter $$AB$$ intersects $$BC$$ at point $$D$$. Connect $$OD$$ and $$AD$$. The area of the shaded region in the figure is         ."},{"type":"image_path","image_path":"images/2405_q0.png"}],"answer":"$$4 \\pi - 8$$"}
{"id":"2416","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, angles $$\\angle B$$ and $$\\angle C$$ are folded as shown in the figure, with points B and C both landing at a point G on side $$B C$$, and segments $$M N$$, $$E F$$ are the fold lines. If $$\\angle A = 94 \\circ$$, then $$\\angle M G E =$$      ."},{"type":"image_path","image_path":"images/2416_q0.png"}],"answer":"$$94 \\circ$$"}
{"id":"2419","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, given AB ∥ DE, ∠ABC = 75°, ∠CDE = 150°, find the measure of ∠BCD."},{"type":"image_path","image_path":"images/2419_q0.png"}],"answer":"45°"}
{"id":"2437","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in right triangle $$\\triangle$$ABC, ∠B=90°, ED is the perpendicular bisector of AC, intersecting AC at point D and BC at point E. Given that ∠BAE=10°, the measure of ∠C is             °."},{"type":"image_path","image_path":"images/2437_q0.png"}],"answer":"40°"}
{"id":"2500","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A = 50 \\circ$$. It is folded such that point A lands at point $$A^{'}$$ on side $$C B$$, with the crease being $$C D$$. Find the measure of $$\\angle A^{'} D B$$."},{"type":"image_path","image_path":"images/2500_q0.png"}],"answer":"$$10 \\circ$$"}
{"id":"2511","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A D$$ is the median of $$\\triangle A B C$$, points E and F are the midpoints of $$A D$$ and $$A C$$ respectively, connect $$E F$$, if $$E F = 3$$, then the length of $$A D$$ is        ."},{"type":"image_path","image_path":"images/2511_q0.png"}],"answer":"$$6$$"}
{"id":"2532","difficulty":"0.8","question_list":[{"type":"text","text":"In the ancient Chinese mathematical text \"The Nine Chapters on the Mathematical Art,\" there is the following problem: \"A bamboo stick is 1 zhang tall. After it breaks, the top of the bamboo falls 3 feet away from the base of the bamboo. How high above the ground is the break point?\" (Note: 1 zhang = 10 feet). As shown in the figure, according to the problem, let the top of the broken bamboo fall at point A, the base of the bamboo be point B, and the break point be point C. The length of BC, the height of the break point from the ground, is              feet."},{"type":"image_path","image_path":"images/2532_q0.png"}],"answer":"4.55"}
{"id":"2535","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle paper $$ABC$$, $$\\angle ACB = 90^\\circ$$, $$AC = 6$$, $$BC = 8$$. Fold the paper so that point B coincides with point A, then unfold to obtain the crease $$DE$$. The length of $$CD$$ is      ."},{"type":"image_path","image_path":"images/2535_q0.png"}],"answer":"$$\\frac{7}{4}$$"}
{"id":"2545","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 2. $$E$$ is a moving point that does not coincide with point $$D$$; construct square $$D E F G$$ with $$D E$$ as one side. Let $$D E = d_{1}$$, and the distances from points $$F$$ and $$G$$ to point $$C$$ are $$d_{2}$$ and $$d_{3}$$, respectively. Then the minimum value of $$d_{1} + d_{2} + d_{3}$$ is        ."},{"type":"image_path","image_path":"images/2545_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"2645","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, to cover a staircase that is 5 m high and 13 m long with red carpet, at least      m of red carpet is needed."},{"type":"image_path","image_path":"images/2645_q0.png"}],"answer":"17"}
{"id":"2663","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 6$$, $$B C = 8$$, point $$M$$ is the midpoint of $$A B$$, find $$C M =$$       ."},{"type":"image_path","image_path":"images/2663_q0.png"}],"answer":"5"}
{"id":"2684","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle O A_{1} A_{2}$$ is an isosceles right triangle, $$O A_{1} = 1$$, and an isosceles right triangle $$\\text{Rt} \\triangle O A_{2} A_{3}$$ is constructed with hypotenuse $$O A_{2}$$ as one leg; then an isosceles right triangle $$\\text{Rt} \\triangle O A_{3} A_{4}$$ is constructed with $$O A_{3}$$ as one leg, and so on. Following this pattern, a spiral pattern is obtained. The length of $$O A_{n}$$ is        .  (Expressed in terms of n)"},{"type":"image_path","image_path":"images/2684_q0.png"}],"answer":"$$\\left(\\sqrt{2}\\right)^{n - 1}$$"}
{"id":"2696","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, $$BD$$ is the diagonal of rectangle $$ABCD$$, $$BE$$ bisects $$\\angle ABD$$, if $$\\angle BDC = 50^{\\circ}$$, then $$\\angle AEB =$$       °."},{"type":"image_path","image_path":"images/2696_q0.png"}],"answer":"$$65$$"}
{"id":"2770","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square ABCD$$, $$AC$$ is a diagonal, $$\\angle ACD = 90^\\circ$$, E is the midpoint of $$BC$$, $$AF$$ bisects $$\\angle BAC$$, and connect $$CF$$, $$EF$$. If $$CF \\perp AF$$, $$AB = 5$$, $$BC = 13$$, then the length of $$EF$$ is       ."},{"type":"image_path","image_path":"images/2770_q0.png"}],"answer":"$$\\frac{7}{2}$$/3.5"}
{"id":"2775","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$\\angle B = 62 \\circ$$, D and E are on $$A B$$ and $$A C$$ respectively. Folding $$\\triangle A D E$$ along $$D E$$ gives $$\\triangle F D E$$, and it satisfies $$E F \\parallel A B$$. Then $$\\angle 1 =$$        ."},{"type":"image_path","image_path":"images/2775_q0.png"}],"answer":"$$76 \\circ$$"}
{"id":"2786","difficulty":"0.8","question_list":[{"type":"text","text":"As shown in the figure, which is formed by combining a set of triangle rulers, the degree measure of ∠ABC in the figure is        ."},{"type":"image_path","image_path":"images/2786_q0.png"}],"answer":"75°"}
{"id":"2804","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B = 90 \\circ$$, point $$E$$ is a moving point on side $$B C$$, $$D C \\bot B C$$, connect $$A E$$, $$D E$$. $$D E$$ intersects $$A C$$ at point $$F$$, $$\\angle D F C = 45 \\circ$$, $$A C = 2 \\sqrt{10}$$, $$C E = 3 \\sqrt{2}$$, if $$B E = D C$$, then $$A E$$      ."},{"type":"image_path","image_path":"images/2804_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"2806","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$AC = BC$$, $$\\angle C = 30^\\circ$$, $$BD$$ bisects $$\\angle ABC$$ and intersects $$AC$$ at $$D$$, points $$E$$ and $$F$$ are on $$AB$$ and $$BC$$ respectively. When $$\\triangle BEF$$ is folded along $$EF$$, point $$B$$ coincides exactly with point $$D$$. If the perimeter of quadrilateral $$BEDF$$ is 16, then the length of $$CD$$ is           ."},{"type":"image_path","image_path":"images/2806_q0.png"}],"answer":"$$2 \\sqrt{6} + 2 \\sqrt{2}$$/$$2 \\sqrt{2} + 2 \\sqrt{6}$$"}
{"id":"2814","difficulty":"0.4","question_list":[{"type":"text","text":"In Rt△ACB, ∠ACB=90°, point D is on side AB, connect CD, fold △ADC along the straight line CD, point A happens to fall on point E on side BC, if AC=3, BE=1, then the length of DE is     ."},{"type":"image_path","image_path":"images/2814_q0.png"}],"answer":"$$\\frac{15}{7}$$"}
{"id":"2820","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, this is a schematic diagram of the \"string figure\" used by the ancient Chinese mathematician Zhao Shuang to prove the Pythagorean theorem. It consists of four congruent right triangles and a small square $$EFGH$$, exactly forming a large square $$ABCD$$. Connect $$EG$$ and extend it to intersect $$BC$$ at point $$M$$. If $$AB = 5$$ and $$EF = 1$$, then the length of $$GM$$ is        ."},{"type":"image_path","image_path":"images/2820_q0.png"}],"answer":"$$\\frac{9 \\sqrt{2}}{7}$$"}
{"id":"2826","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = 10, t a n A = 2, B E \\bot A C$$ at point E, and D is a moving point on segment $$B E$$, then the minimum value of $$\\sqrt{5} C D + B D$$ is          ."},{"type":"image_path","image_path":"images/2826_q0.png"}],"answer":"20"}
{"id":"2834","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the square $$ABCD$$ has an area of 16, and its diagonals $$AC$$, $$BD$$ intersect at point $$O$$. Points $$E$$ and $$F$$ move along sides $$AB$$ and $$BC$$ respectively, with $$\\angle EOF = 90^\\circ$$, and $$OG$$ bisects $$\\angle EOF$$, intersecting side $$BC$$ at point $$G$$. The following conclusions are given:\n① $$OE = OF$$;\n② The area of quadrilateral $$OEBF$$ remains constant at 4;\n③ $$BG^{2} + CF^{2} = GF^{2}$$;\n④ The minimum value of $$EF$$ is $$2\\sqrt{2}$$.\nWhich of the above statements are correct? (Fill in all the correct statement numbers)"},{"type":"image_path","image_path":"images/2834_q0.png"}],"answer":"①②③④"}
{"id":"2853","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the circle $$\\bigodot O$$ with radius 3, B is the midpoint of the minor arc AC. Connect AB and extend it to D such that $$BD = AB$$. Connect AC, BC, and CD. If $$AB = 2$$, then CD equals      ."},{"type":"image_path","image_path":"images/2853_q0.png"}],"answer":"$$\\frac{4}{3}$$"}
{"id":"2903","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$ABC$$, $$AB = BC = 4\\sqrt{2}$$, point $$P$$ lies on the circle with diameter $$AC$$, and $$M$$ is the midpoint of $$PB$$. When point $$P$$ moves along the circle from point $$A$$ through one full revolution, the minimum value of the length of $$CM$$ is      ."},{"type":"image_path","image_path":"images/2903_q0.png"}],"answer":"$$2 \\sqrt{5} - 2$$"}
{"id":"2904","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, the sides of ∠EAF intersect the extensions of CB and DC at points E and F respectively, and ∠EAF = 45°. If BE = 1 and DF = 7, then EF =  ."},{"type":"image_path","image_path":"images/2904_q0.png"}],"answer":"6"}
{"id":"2907","difficulty":"0.4","question_list":[{"type":"text","text":"In the rhombus $$A B C D$$, $$A B = 4$$, $$\\angle B = 2 \\angle A$$, points E and F are the midpoints of $$A D$$ and $$A B$$ respectively. A moving point P starts from B and moves clockwise to point C. When $$\\triangle P E F$$ is a right triangle, the length of $$B P$$ is      ."},{"type":"image_path","image_path":"images/2907_q0.png"}],"answer":"3 or $$\\sqrt{13}$$ or $$2 \\sqrt{3}$$"}
{"id":"2916","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point E is a moving point on the diagonal AC of rectangle ABCD. From point E, draw EF ⊥ BC, intersecting at point F. Given AB = 3 and BC = 6, if there is a point G on AB such that triangle EFG is an isosceles right triangle, then the length of AG is ______."},{"type":"image_path","image_path":"images/2916_q0.png"}],"answer":"3 or 1 or $$\\frac{9}{5}$$"}
{"id":"2923","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $ABCD$ with side length $4$, point $E$ is the midpoint of side $BC$, and point $F$ is a moving point on side $AB$. An equilateral triangle $FEG$ is constructed with $EF$ as one side, located above and to the right of $EF$. When $CG$ is minimized, the perimeter of $\\triangle ECG$ is      ."},{"type":"image_path","image_path":"images/2923_q0.png"}],"answer":"$$5 + \\sqrt{7}$$/$$\\sqrt{7} + 5$$"}
{"id":"2936","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ, \\angle B = 30 \\circ$$.D is a moving point on side $$B C$$, and connect $$A D$$.When $$A D + \\frac{1}{2} B D$$ takes its minimum value, the value of $$\\frac{B D}{B C}$$ is        ."},{"type":"image_path","image_path":"images/2936_q0.png"}],"answer":"$$\\frac{2}{3}$$"}
{"id":"2946","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C$$, draw a line $$l$$ through point $$C$$ parallel to $$A B$$, and let point $$P$$ be a moving point on line $$l$$ different from point $$C$$. Connect $$A P$$, and from point $$P$$ draw a perpendicular to $$A P$$ intersecting line $$B C$$ at point $$D$$. If $$A C = 7 \\sqrt{2}$$, $$A P = 25$$, then the length of segment $$B D$$ is       ."},{"type":"image_path","image_path":"images/2946_q0.png"}],"answer":"$$17 \\sqrt{2}$$ or $$31 \\sqrt{2}$$"}
{"id":"2951","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the diameter AB of ⊙O is 2, and C is a fixed point on ⊙O, with ∠ABC = 30°. A moving point P starts from A and moves along the semicircular arc $$\\hat{AB}$$ toward point B (point P and point C lie on opposite sides of diameter AB). The motion stops when P reaches point B. During the motion, from point C, draw CD perpendicular to CP, intersecting the extension of PB at point D, and connect AD. The maximum value of segment AD is                   ."},{"type":"image_path","image_path":"images/2951_q0.png"}],"answer":"$$\\sqrt{7} + \\sqrt{3}$$/$$\\sqrt{3} + \\sqrt{7}$$"}
{"id":"2967","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, AP and BP bisect the interior angle ∠CAB and the exterior angle ∠CBD of △ABC, respectively. Connect CP. If ∠ACP = 130°, then ∠APB =   ."},{"type":"image_path","image_path":"images/2967_q0.png"}],"answer":"$$40 \\circ$$"}
{"id":"2973","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 7$$, F is a moving point on side $$CD$$. Rotate $$\\triangle ADF$$ clockwise about point A by $$90^\\circ$$ to obtain $$\\triangle ABE$$, and reflect $$\\triangle ADF$$ across line AF to obtain $$\\triangle AGF$$. Let EF and BD intersect at point H, and connect GH. Then the maximum area of $$\\triangle EGH$$ is       ."},{"type":"image_path","image_path":"images/2973_q0.png"}],"answer":"$$\\frac{49}{16}$$"}
{"id":"2974","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the equilateral triangle $$\\triangle A B C$$ has side length 4, and the radius of $$\\bigodot C$$ is 2. Point P is a moving point on $$A B$$, and from point P, a tangent $$P Q$$ is drawn to $$\\bigodot C$$, with Q being the point of tangency. Then the minimum value of $$P Q$$ is                ."},{"type":"image_path","image_path":"images/2974_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"2983","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$A D = 8$$, $$E$$ is a moving point on $$B C$$ (not coinciding with points $$B$$, $$C$$). Connect $$A E$$, and from point $$D$$, draw $$D F \\bot A E$$, with foot of perpendicular at $$F$$. Then the minimum length of segment $$B F$$ is     ."},{"type":"image_path","image_path":"images/2983_q0.png"}],"answer":"$$2 \\sqrt{13} - 4$$"}
{"id":"2988","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠BAC = ∠BCA = 44°, M is a point inside △ABC; and ∠MCA = 30°, ∠MAC = 16°, then the measure of ∠BMC is    ."},{"type":"image_path","image_path":"images/2988_q0.png"}],"answer":"150°"}
{"id":"3005","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$ABCD$$, $$AB \\parallel CD$$, $$\\angle B = \\angle C = 90^\\circ$$, $$\\triangle ADE$$ is an equilateral triangle, and point $$E$$ lies on $$BC$$. If $$AB = 6$$, $$CD = 9$$, the area of $$\\triangle ADE$$ is        ."},{"type":"image_path","image_path":"images/3005_q0.png"}],"answer":"$$21 \\sqrt{3}$$"}
{"id":"3027","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, $$A B = 6$$, $$P$$ is a moving point on $$A B$$, $$P D \\bot B C$$, $$P E \\bot A C$$, then the minimum value of $$D E$$ is        ."},{"type":"image_path","image_path":"images/3027_q0.png"}],"answer":"$$\\frac{9}{2}$$"}
{"id":"3049","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, point $$E$$ is a point on side $$A D$$, connect $$B E$$, rotate segment $$E B$$ counterclockwise around point $$E$$ by $$120^{\\circ}$$ to obtain $$E F$$, $$E F$$ intersects $$C D$$ at point $$H$$, connect $$B F$$, which intersects $$C D$$ at point $$G$$. Given $$\\angle A = 120^{\\circ}$$, $$A B = 3$$, $$A E = 1$$, then $$B E =$$    , $$\\frac{F G}{B G}$$=      ."},{"type":"image_path","image_path":"images/3049_q0.png"}],"answer":"$$\\sqrt{13}$$     $$\\frac{2}{3}$$"}
{"id":"3058","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point O is the center of square $$A B C D$$, with $$A B = 3 \\sqrt{2}$$. In $$R t \\triangle B E F$$, $$\\angle B E F = 90^\\circ$$, and $$E F$$ passes through point D; $$B E$$ and $$B F$$ intersect $$A D$$ and $$C D$$ at points G and M, respectively. Connect $$O E$$, $$O M$$, and $$E M$$. If $$B G = D F$$ and $$\\tan \\angle A B G = \\frac{1}{3}$$, then the perimeter of $$\\triangle O E M$$ is           ."},{"type":"image_path","image_path":"images/3058_q0.png"}],"answer":"$$3 + 3 \\sqrt{5}$$"}
{"id":"3063","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, D is the midpoint of AC, the angle bisector AE of △ABC intersects BD at point F. If BF:FD = 3:1, and AB + BE = 3$$\\sqrt{3}$$, then the perimeter of △ABC is     ."},{"type":"image_path","image_path":"images/3063_q0.png"}],"answer":"$$5 \\sqrt{3}$$"}
{"id":"3070","difficulty":"0.4","question_list":[{"type":"text","text":"In triangle $$\\triangle A B C$$, $$A C = 8$$, $$A B = \\sqrt{41}$$, and the height from $$A$$ to side $$B C$$, denoted as $$A G = 5$$. Point $$D$$ is a moving point on segment $$A C$$. On $$B C$$, take $$C E = A D$$, and connect $$A E$$ and $$B D$$. Then the minimum value of $$A E + B D$$ is      ."},{"type":"image_path","image_path":"images/3070_q0.png"}],"answer":"13"}
{"id":"3075","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B$$ = 6, $$B C$$ = 8, points $$M$$ and $$N$$ are the midpoints of sides $$A D$$ and $$B C$$, respectively. At a certain moment, point $$E$$ starts from point $$M$$ and moves uniformly toward point $$A$$ along the direction $$M A$$ at a speed of 2 units per second; simultaneously, point $$F$$ starts from point $$N$$ and moves uniformly toward point $$C$$ along the direction $$N C$$ at a speed of 1 unit per second. When one of the points reaches a vertex of the rectangle, both points stop moving. Connect $$E F$$, and from point $$B$$, draw a perpendicular to $$E F$$, with foot of the perpendicular at $$H$$. During this motion, the length of the path traced by point $$H$$ is     ."},{"type":"image_path","image_path":"images/3075_q0.png"}],"answer":"$$\\frac{\\sqrt{5}}{2} \\pi$$/$$\\frac{\\sqrt{5} \\pi}{2}$$"}
{"id":"3090","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$\\angle B = 30^{\\circ}$$, $$A C = 2$$, point $$D$$ is the midpoint of $$B C$$, and point $$E$$ is a moving point on side $$A B$$. Fold $$\\triangle B D E$$ along the line $$D E$$ to the position $$\\triangle B^{'} D E$$, with $$B^{'} D$$ intersecting $$A B$$ at point $$F$$, and connect $$A B^{'}$$."},{"type":"image_path","image_path":"images/3090_q0.png"},{"type":"text","text":"(1) The minimum value of $$A B^{'}$$ is        ; (2) If $$\\triangle A B^{'} F$$ is a right triangle, then the length of $$B E$$ is        ."}],"answer":"$$\\sqrt{7} - \\sqrt{3}$$     1 or $$\\frac{6}{5}$$"}
{"id":"3141","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A D \\parallel B C$$, a circle passing through points A, C, and D intersects $$B C$$ at point E. Connect $$A E$$. It is given that $$\\overset{⌢}{A E} = \\overset{⌢}{C E}$$, $$\\angle B A E = \\angle B E A$$. If $$B E = \\frac{5}{6} C E = \\frac{25}{6}$$, then the radius of the circle is        ."},{"type":"image_path","image_path":"images/3141_q0.png"}],"answer":"$$\\frac{5 \\sqrt{5}}{2}$$/$$\\frac{5}{2} \\sqrt{5}$$"}
{"id":"3150","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\bigodot O$$ is the circumcircle of $$\\triangle A B C$$, $$A B = A C$$, $$C D \\bot A B$$ at point $$D$$, the extension of $$B O$$ intersects $$C D$$ at point $$E$$ and intersects $$\\bigodot O$$ at point $$F$$."},{"type":"image_path","image_path":"images/3150_q0.png"},{"type":"text","text":"(1) Let $$\\angle B A C = \\alpha$$, then express $$\\angle D C F =$$      in terms of $$\\alpha$$.  \n(2) If $$B C = 4 \\sqrt{2}$$, $$B E = 4$$, then $$O E =$$      ."}],"answer":"$$90 \\circ - \\frac{1}{2} \\alpha$$     $$1$$"}
{"id":"3157","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$B C = 6$$, $$E$$ and $$F$$ are the midpoints of $$A B$$ and $$A C$$ respectively, point $$P$$ moves along ray $$E F$$, $$B P$$ intersects $$C E$$ at $$D$$, the angle bisector of $$\\angle C B P$$ intersects $$C E$$ at $$Q$$, when $$C Q = \\frac{1}{2} C E$$, $$E P + B P =$$       ."},{"type":"image_path","image_path":"images/3157_q0.png"}],"answer":"$$12$$"}
{"id":"3164","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle A B C = 60 \\circ$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, and $$P$$ is a moving point on diagonal $$B D$$, with $$P M \\bot A B$$ at point $$M$$ and $$P N \\bot A D$$ at point $$N$$. The following conclusions are given: ① $$\\triangle A B C$$ is an equilateral triangle, ② $$O B = \\sqrt{3} O A$$, ③ $$\\angle M P N = 60 \\circ$$, ④ $$P M + P N = \\frac{1}{2} B D$$. Among these, the correct ones are         (fill in the serial numbers)"},{"type":"image_path","image_path":"images/3164_q0.png"}],"answer":"①②③④"}
{"id":"3182","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a rhombus, $$A B = 4$$, $$\\angle A B C = 60 \\circ$$, $$E$$ and $$F$$ are two moving points on $$B D$$, and $$B E + D F = E F$$, point $$M$$ is the midpoint of $$A D$$, connect $$C E$$ and $$M F$$, then the minimum value of $$C E + M F$$ is      ."},{"type":"image_path","image_path":"images/3182_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"3194","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the angle bisectors BP and AP of ∠ABC and ∠EAC intersect at point P. PM ⊥ BE and PN ⊥ BF, with feet of perpendiculars at M and N, respectively. There are four conclusions:\n① CP bisects ∠ACF; ② ∠BPC = $$\\frac{1}{2}$$∠BAC; ③ ∠APC = 90° − $$\\frac{1}{2}$$∠ABC; ④ S△APM + S△CPN > S△APC.\nAmong these, the correct conclusions are     . (fill in the conclusion numbers)"},{"type":"image_path","image_path":"images/3194_q0.png"}],"answer":"①②③"}
{"id":"3196","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\angle A O B = 45 \\circ$$, there are two moving points C and D on sides $$O A$$ and $$O B$$ respectively. Connect $$C D$$, and construct an isosceles right triangle $$C D E$$ with $$C D$$ as a leg. When the length of $$C D$$ remains constant and equals $$2 \\text{cm}$$, the maximum value of $$O E$$ is                    ."},{"type":"image_path","image_path":"images/3196_q0.png"}],"answer":"$$\\sqrt{2} + \\sqrt{10}$$"}
{"id":"3219","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A D = A B = B C$$, $$A C \\bot B D$$ intersecting at point $$O$$, from vertex $$A$$ of quadrilateral $$A B C D$$, draw $$A F \\bot A D$$, and $$A D = A F$$, segment $$A F$$ intersects $$O B$$ at point $$H$$ and intersects $$B C$$ at point $$E$$. If points $$D$$, $$C$$, $$F$$ are collinear, then the following statements: $$\\textcircled{1}$$ quadrilateral $$A B C D$$ is a rhombus; $$\\textcircled{2}$$ $$A B \\cdot A E = A C \\cdot B D$$; $$\\textcircled{3}$$ $$\\angle A H D = \\angle C A D$$; $$\\textcircled{4}$$ $$O H + O C = \\frac{3}{4} B H$$, the correct ones are      ."},{"type":"image_path","image_path":"images/3219_q0.png"}],"answer":"$$\\textcircled{1} \\textcircled{3}$$"}
{"id":"3224","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, AD = AB, DC = BC, ∠DAB = 60°, ∠DCB = 120°, E is on AD, F is a point on the extension of AB, and DE = BF. If G is on AB and ∠ECG = 60°, then the quantitative relationship among DE, EG, and BG is     ."},{"type":"image_path","image_path":"images/3224_q0.png"}],"answer":"DE+BG=EG"}
{"id":"3242","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rectangle $$A B C D$$ (opposite sides of a rectangle are equal, each angle is $$90 \\circ$$), $$A B = 6 \\text{cm}$$, $$A D = 2 \\text{cm}$$, point $$P$$ and point $$Q$$ start simultaneously from points $$A$$ and $$C$$, respectively. Point $$P$$ moves toward endpoint $$B$$ at a speed of 2 cm/s, and point $$Q$$ moves toward $$D$$ at a speed of 1 cm/s. When one point reaches its endpoint, the other point also stops moving. Let the time of motion be $$t$$."},{"type":"image_path","image_path":"images/3242_q0.png"},{"type":"text","text":"(1) When the distance between points $$P$$ and $$Q$$ is $$3 \\text{cm}$$, $$t =$$          . (2) When $$t =$$       , $$\\triangle P Q D$$ is a right triangle ($$t \\neq 0$$)."}],"answer":"$$t = \\frac{6 - \\sqrt{5}}{3}$$ or $$\\frac{6 + \\sqrt{5}}{3}$$     $$t = \\frac{3 + \\sqrt{3}}{3}$$ or $$\\frac{3 - \\sqrt{3}}{3}$$ or 2"}
{"id":"3259","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, points $$D,  E,  F$$ are on sides $$A B,  B C,  A C$$ respectively, and $$A E,  B F,  C D$$ intersect at point $$G$$. If $$\\frac{A G}{G E} + \\frac{B G}{G F} + \\frac{C G}{G D} = 2014$$, then the value of $$\\frac{A G}{G E} \\cdot \\frac{B G}{G F} \\cdot \\frac{C G}{G D}$$ is            ."},{"type":"image_path","image_path":"images/3259_q0.png"}],"answer":"2016"}
{"id":"3261","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, square $$ABCD$$, point $$E$$ is a moving point on ray $$AB$$, draw $$EF \\parallel DB$$, intersecting line $$AD$$ at point $$F$$, connect $$DE$$, take the midpoint $$G$$ of $$DE$$, connect $$FG$$ and extend it to intersect line $$DB$$ at point $$H$$. If $$AB = 4$$, $$EB = 3$$, then the length of $$FH$$ is      ."},{"type":"image_path","image_path":"images/3261_q0.png"}],"answer":"$$\\sqrt{5}$$ or $$\\sqrt{149}$$/$$\\sqrt{149}$$ or $$\\sqrt{5}$$"}
{"id":"3279","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 8, point E is on $$D C$$, point F is on $$A C$$, $$\\angle B F E = 90 \\circ$$, and if $$C E = 2$$, then the length of $$A F$$ is      ."},{"type":"image_path","image_path":"images/3279_q0.png"}],"answer":"$$\\text{3} \\sqrt{\\text{2}}$$"}
{"id":"3281","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = AC, BD ⊥ AC at point D. Rotate segment AC about point C to obtain segment CE, such that point E lies exactly on the extension of AB. $$BE = \\frac{1}{2} CD$$, and the area of △BCD is 8. Find the length of BC."},{"type":"image_path","image_path":"images/3281_q0.png"}],"answer":"$$2 \\sqrt{10}$$"}
{"id":"3282","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B - A C = 4$$, $$B C = 7$$, $$B D$$ is perpendicular to the angle bisector $$A D$$ of $$\\angle B A C$$ at point $$D$$, $$E$$ is the midpoint of $$A C$$, and $$B E$$ intersects $$A D$$ at $$F$$. Find the maximum value of the difference in areas of $$\\triangle B D F$$ and $$\\triangle A E F$$."},{"type":"image_path","image_path":"images/3282_q0.png"}],"answer":"$$7$$"}
{"id":"3313","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$A D \\bot B C$$ at point $$D$$, $$D E \\bot A B$$ at point $$E$$. If $$\\angle B = 30 \\circ$$, $$A E = 1$$."},{"type":"image_path","image_path":"images/3313_q0.png"},{"type":"text","text":"(1) The length of $$B E$$ is      ; (2) Take a point $$M$$ on one of the legs of $$\\triangle A B C$$, when $$\\triangle D E M$$ is an isosceles triangle, the length of $$B M$$ is      ."}],"answer":"$$3$$     $$3 - \\sqrt{3}$$ or $$\\sqrt{21}$$"}
{"id":"3322","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, point D is the midpoint of the hypotenuse $$A B$$, from point B, draw $$B E \\bot C D$$ intersecting at F, intersecting $$A C$$ at point E, from point A, draw $$A G \\parallel B C$$, intersecting the extension of $$B E$$ at point G. If $$A E = B C$$, then the value of $$\\frac{D F}{C F}$$ is      ."},{"type":"image_path","image_path":"images/3322_q0.png"}],"answer":"$$\\frac{\\sqrt{5} + 1}{4}$$"}
{"id":"3332","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, it is given that $$\\angle A C B = 90^{\\circ}$$, $$A C = 4$$, $$B C = 2 \\sqrt{5}$$, points $$D$$ and $$E$$ are two moving points on sides $$A B$$ and $$A C$$ respectively, and $$A D = C E$$ always holds. Connect $$B E$$ and $$C D$$. When $$C D + B E$$ takes its minimum value, $$A E =$$           ."},{"type":"image_path","image_path":"images/3332_q0.png"}],"answer":"$$\\frac{12}{5}$$"}
{"id":"3347","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, the angle bisector of $$\\angle DAB$$ intersects the external angle bisector of $$\\angle ABC$$ at point P, and $$\\angle D + \\angle C = 240^\\circ$$. Then $$\\angle P =$$      ."},{"type":"image_path","image_path":"images/3347_q0.png"}],"answer":"$$30 \\circ$$"}
{"id":"3349","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$ with side length 4, D and E are the midpoints of $$A B$$ and $$B C$$ respectively. Connect $$D E$$, and let F be the midpoint of $$D E$$. Connect $$A F$$. Then the length of $$A F$$ is         ."},{"type":"image_path","image_path":"images/3349_q0.png"}],"answer":"$$\\sqrt{7}$$"}
{"id":"3356","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B =\\text{60} \\circ$$, rotate $$\\triangle A B C$$ clockwise about point C to the position of $$\\triangle C D E$$, let F be a point on CE, connect AF and extend it to intersect the extension of BC at point G. If CF bisects $$\\angle A C G$$, $$\\angle C A F = 2 \\angle E A F$$, $$\\angle A E D = 21 \\circ$$, then $$\\angle G$$ is      °."},{"type":"image_path","image_path":"images/3356_q0.png"}],"answer":"$$34$$"}
{"id":"3374","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, $$A B = 8$$, $$B C = 10$$, diagonal $$B D = 6$$, P is a moving point on $$A D$$, and Q is a fixed point on $$A B$$. The minimum value of $$3 D P + 5 P Q$$ is         ."},{"type":"image_path","image_path":"images/3374_q0.png"}],"answer":"30"}
{"id":"3393","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, ∠C + ∠D = $$210 \\circ$$, E and F are points on AD and BC respectively. Quadrilateral CDEF is folded along line EF to obtain quadrilateral $$C^{'} D^{'} E F$$, $$C^{'} F$$ intersects AD at point G. If △EFG has two equal angles, then ∠EFG =          ."},{"type":"image_path","image_path":"images/3393_q0.png"}],"answer":"$$40 \\circ$$ or $$50 \\circ$$"}
{"id":"3408","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$A D = 4$$, point $$E$$ is the midpoint of $$A D$$, point $$F$$ lies on line $$C D$$, point $$G$$ lies on segment $$E F$$, and $$\\angle G D F = \\angle D E F$$, point $$P$$ is a moving point on side $$B C$$, then the minimum value of $$P A + P G$$ is      ."},{"type":"image_path","image_path":"images/3408_q0.png"}],"answer":"$$3 \\sqrt{5} - 1$$/$$- 1 + 3 \\sqrt{5}$$"}
{"id":"3411","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$B D$$ is the median to side $$A C$$, and $$A E$$ is the median to side $$B D$$ in $$\\triangle A B D$$. If $$\\angle C B D = 60 \\circ$$, $$\\angle A E B = 150 \\circ$$, and $$B D = 4$$, then $$A B =$$          ."},{"type":"image_path","image_path":"images/3411_q0.png"}],"answer":"$$7$$"}
{"id":"3420","difficulty":"0.4","question_list":[{"type":"text","text":"In △ABC, $$\\angle A C B = 90 \\circ, \\angle B = 60 \\circ,$$ $$A B = 4,$$ point D is a moving point on line BC, connect AD, construct an equilateral △ADE on the right side of line AD, connect CE, when the length of segment CE is minimized, the length of segment $$C D$$ is   ."},{"type":"image_path","image_path":"images/3420_q0.png"}],"answer":"$$3$$"}
{"id":"3430","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is $$4$$, and points $$E$$ and $$F$$ are on sides $$B C$$ and $$C D$$ respectively, with $$C E = D F$$. Lines $$A F$$ and $$D E$$ intersect at point $$O$$, and $$B O = B A$$. Find the value of $$O C$$."},{"type":"image_path","image_path":"images/3430_q0.png"}],"answer":"$$\\frac{4 \\sqrt{10}}{5}$$"}
{"id":"3434","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given that the side length of square $$A B C D$$ is $$3$$, and $$E$$, $$F$$ are points on sides $$A B$$ and $$B C$$ respectively, with $$\\angle E D F = 45^\\circ$$. Rotate $$\\triangle D A E$$ counterclockwise about point $$D$$ by $$90^\\circ$$ to obtain $$\\triangle D C M$$. If $$A E = 1$$, then the length of $$F M$$ is  ."},{"type":"image_path","image_path":"images/3434_q0.png"}],"answer":"$$2.5$$"}
{"id":"3463","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, $$AB = 2$$, points C and D are on the same side of $$AB$$, and $$AD \\bot AB$$, $$BC \\bot AB$$, $$AD = 1$$, $$BC = 3$$. Point P is a moving point on $$\\bigodot O$$. Find the minimum value of $$\\frac{\\sqrt{2}}{2} PD + PC$$ ."},{"type":"image_path","image_path":"images/3463_q0.png"}],"answer":"$$\\frac{\\sqrt{34}}{2}$$"}
{"id":"3464","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$B E = 6$$, $$\\angle B = 60 \\circ$$, point $$P$$ is a moving point on segment $$A C$$, and point $$F$$ is a moving point on segment $$A B$$. When $$E P + F P$$ is minimized, connect $$E F = 6$$, then the length of $$A B$$ is         ."},{"type":"image_path","image_path":"images/3464_q0.png"}],"answer":"$$\\text{18}$$"}
{"id":"3468","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$ABCD$$, $$BC = 2AB$$, point P is a moving point on side $$AD$$, and an equilateral triangle $$\\triangle BPP'$$ is constructed to the right with $$BP$$ as its side. Connect $$CP'$$. When point $$P'$$ falls on side $$BC$$, the measure of $$\\angle PP'C$$ is          ; when the length of segment $$CP'$$ is minimized, the measure of $$\\angle PP'C$$ is         ."},{"type":"image_path","image_path":"images/3468_q0.png"}],"answer":"$$120 \\circ$$     $$75 \\circ$$"}
{"id":"3476","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given $$\\angle M O N$$, take points $$P_{1}$$, $$P_{3}$$, $$P_{5} \\ldots$$ successively on side $$O N$$, and points $$P_{2}$$, $$P_{4}$$, $$P_{6} \\ldots$$ successively on side $$O M$$, such that $$O P_{1} = P_{1} P_{2} = P_{2} P_{3} = P_{3} P_{4} = P_{4} P_{5} \\ldots$$, forming isosceles triangles $$\\triangle O P_{1} P_{2}$$, $$\\triangle P_{1} P_{2} P_{3}$$, $$\\triangle P_{2} P_{3} P_{4}$$, $$\\triangle P_{3} P_{4} P_{5} \\ldots$$"},{"type":"image_path","image_path":"images/3476_q0.png"},{"type":"text","text":"(1) If $$\\angle M O N = 30^\\circ$$, the last isosceles triangle obtained is            ; (2) If the above procedure is followed, and the last isosceles triangle obtained is $$\\triangle P_{3} P_{4} P_{5}$$, then the range of possible values for $$\\angle M O N$$, denoted as $$\\alpha$$, is            ."}],"answer":"△$$P_{1} P_{2} P_{3}$$     $$18 \\circ \\leq \\alpha < 22.5 \\circ$$"}
{"id":"3482","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A C B$$, $$\\angle A C B = 90 \\circ$$, $$\\angle B = 30 \\circ$$, $$B C = 4 \\sqrt{3}$$, draw an arc with center at point C and radius equal to $$A C$$, intersecting $$A B$$ and $$B C$$ at points D and E respectively; draw another arc with center at point E and radius equal to $$C E$$, intersecting $$A B$$ at point F and intersecting $$\\overset{⌢}{A E}$$ at point G. The area of the shaded region in the figure is          ."},{"type":"image_path","image_path":"images/3482_q0.png"}],"answer":"$$4 \\pi - 4 \\sqrt{3}$$"}
{"id":"3489","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, AD = 4, CD = 3, ∠ABC = ∠ACB = ∠ADC = 45°, then the length of BD is       ."},{"type":"image_path","image_path":"images/3489_q0.png"}],"answer":"$$\\sqrt{41}$$"}
{"id":"3494","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in sector $$OAB$$, $$\\angle AOB = 120^\\circ$$, radius $$OA = 2$$, point C is a point on $$OA$$, connect $$BC$$, fold the sector along $$BC$$ such that point A lands at point D on the extension of $$BO$$, connect $$CD$$, then the area of the shaded region in the figure is          (answer in terms of $$\\pi$$)."},{"type":"image_path","image_path":"images/3494_q0.png"}],"answer":"$$\\frac{4}{3} \\pi - 3 + \\sqrt{3}$$"}
{"id":"3497","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C = 3$$, $$\\angle C = 90 \\circ$$, point D lies on side $$B C$$ (not coinciding with points B or C), connect $$A D$$, point E lies on side $$A B$$, $$\\angle E D B = \\angle A D C$$. It is known that point H lies on ray $$A C$$, connect $$E H$$ intersecting segment $$A D$$ at point G. When $$C H = 1$$, and $$\\angle A E H = \\angle B E D$$, then $$\\frac{B E}{A B} =$$        ."},{"type":"image_path","image_path":"images/3497_q0.png"}],"answer":"$$\\frac{1}{2}$$ or $$\\frac{1}{5}$$"}
{"id":"3512","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the angle formed by lines $$l_{1}$$ and $$l_{2}$$, $$\\angle B_{1} O A_{1} = 30 \\circ$$, draw $$A_{1} B_{1} \\bot l_{1}$$ intersecting line $$l_{2}$$ at point $$B_{1}$$, with $$O B_{1} = 2$$. Construct an equilateral triangle $$A_{1} B_{1} C_{1}$$ with $$A_{1} B_{1}$$ as a side outside $$\\triangle O A_{1} B_{1}$$. Then, from point $$C_{1}$$, draw $$A_{2} B_{2} \\bot l_{1}$$, intersecting lines $$l_{1}$$ and $$l_{2}$$ at points $$A_{2}$$ and $$B_{2}$$ respectively. Construct an equilateral triangle $$A_{2} B_{2} C_{2}$$ with $$A_{2} B_{2}$$ as a side outside $$\\triangle O A_{2} B_{2}$$, and so on. Following this pattern, the perimeter of the 2023rd equilateral triangle $$A_{2023} B_{2023} C_{2023}$$ is           ."},{"type":"image_path","image_path":"images/3512_q0.png"}],"answer":"$$\\frac{3^{2023}}{2^{2022}}$$"}
{"id":"3534","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$B E$$ bisects $$\\angle A B C$$, $$\\angle B O D = 45 \\circ$$, $$O F \\bot A D$$, the following conclusions: ① $$A D$$ bisects $$\\angle B A C$$; ② $$A D = O G + O F$$; ③ If $$B D = 3$$, $$A B = 12$$, then $$A G = 9$$; ④ $$S_{\\triangle A C D} : S_{\\triangle A B D} = A B : A C$$. Among these, the correct ones are           (fill in the serial numbers)."},{"type":"image_path","image_path":"images/3534_q0.png"}],"answer":"①②③"}
{"id":"3563","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle M N G$$, $$M N = 4 \\sqrt{2}$$, $$\\angle M = 75 \\circ$$, $$M G = 3$$, point $$O$$ is a point inside $$\\triangle M N G$$, then the minimum value of the sum of the distances from point $$O$$ to the three vertices of $$\\triangle M N G$$ is        ."},{"type":"image_path","image_path":"images/3563_q0.png"}],"answer":"$$\\sqrt{65}$$"}
{"id":"3567","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given square ABCD, point M is a moving point on the extension of side BA (not coinciding with point A), and AM < AB. Triangle CBE is obtained by translating triangle DAM. If from point E, EH is drawn perpendicular to AC, with H as the foot of the perpendicular, then the following conclusions hold:\n① When point M moves to a position such that ∠DHC = 60°, then 2BE = DM;\n② Regardless of where point M moves, DM = √2 HM;\n③ During the motion of point M, quadrilateral CEMD cannot be a rhombus;\n④ Regardless of where point M moves, ∠CHM is always greater than 135°.\nWhich of the above conclusions are correct? (Fill in all the correct conclusion numbers.)"},{"type":"image_path","image_path":"images/3567_q0.png"}],"answer":"①②③④"}
{"id":"3570","difficulty":"0.4","question_list":[{"type":"text","text":"Given the isosceles right triangle $$A B C$$ and the isosceles right triangle $$A D E$$ with right-angle vertex $$A$$ coinciding, $$A B = A C = 3 \\sqrt{2}$$, $$A D = A E = 2$$. Connect $$B E$$ and $$C D$$. Rotate $$\\triangle A D E$$ around point $$A$$ in the plane; the rotated triangle is $$A E^{'} D^{'}$$. If point $$M$$ is the midpoint of $$B E$$, and when points $$E^{'}$$, $$D^{'}$$, and $$C$$ are collinear, the length of segment $$A M$$ is     ."},{"type":"image_path","image_path":"images/3570_q0.png"}],"answer":"$$2 + \\frac{\\sqrt{2}}{2}$$ or $$2 - \\frac{\\sqrt{2}}{2}$$"}
{"id":"3571","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, construct a square $ABDE$ on the hypotenuse $AB$ of the right triangle $ABC$, on the same side as $\\triangle ABC$. Let $O$ be the center of the square, and connect $OC$. If $AB = 13$ and $AC = 5$, then the length of $OC$ is          ."},{"type":"image_path","image_path":"images/3571_q0.png"}],"answer":"$$\\frac{7 \\sqrt{2}}{2}$$"}
{"id":"3576","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in sector $$OAB$$, $$\\angle AOB = 90^\\circ$$, radius $$OA = 8$$. Point $$F$$ is located at $$\\frac{1}{3}$$ of arc $$\\overset{⌢}{AB}$$, closer to point $$A$$. Points $$C$$ and $$D$$ lie on segments $$OA$$ and $$OB$$ respectively, with $$CD = 8$$. $$E$$ is the midpoint of $$CD$$. Connect $$EF$$ and $$BE$$. As $$CD$$ slides (with its length remaining constant), when $$EF$$ takes its minimum value, the perimeter of the shaded region is           ."},{"type":"image_path","image_path":"images/3576_q0.png"}],"answer":"$$4 + 4 \\sqrt{3} + \\frac{8}{3} \\pi$$"}
{"id":"3589","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$C B = 7$$, $$A C = 9$$, a circle $$\\bigodot C$$ with center $$C$$ and radius 3 is drawn. Point $$P$$ is a moving point on $$\\bigodot C$$. Connect $$A P$$ and $$B P$$. The minimum value of $$\\frac{1}{3} A P + B P$$ is      ."},{"type":"image_path","image_path":"images/3589_q0.png"}],"answer":"$$5 \\sqrt{2}$$"}
{"id":"3591","difficulty":"0.4","question_list":[{"type":"text","text":"A triangular piece of material is shown in the figure, $$\\angle B = 30^\\circ, \\angle C = 90^\\circ$$, $$AC = 4$$, cut out a rectangle $$CDEF$$ from this material, where points D, E, F lie on $$BC, AB, AC$$ respectively, the maximum area of the rectangle $$CDEF$$ that can be cut out is      ."},{"type":"image_path","image_path":"images/3591_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"3596","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C$$, $$B D$$ intersect at point O. Points E and F are two moving points on $$O D$$ and $$O C$$ respectively, with $$E F = 4$$. P is the midpoint of $$E F$$. Connect $$O P$$, $$P C$$, $$P D$$. If $$A C = 12$$, $$B D = 16$$, then the minimum value of $$P C + \\frac{1}{4} P D$$ is                ."},{"type":"image_path","image_path":"images/3596_q0.png"}],"answer":"$$\\frac{\\sqrt{145}}{2}$$/$$\\frac{1}{2} \\sqrt{145}$$"}
{"id":"3602","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle B A C = 30 \\circ$$, point $$D$$ is the midpoint of $$A C$$, connect $$B D$$, fold $$\\triangle A B D$$ along $$B D$$ to obtain $$\\triangle A^{'} B D$$, connect $$A A^{'}$$, $$C A^{'}$$, if $$B C = 1$$, then the length of $$A^{'} C$$ is      ."},{"type":"image_path","image_path":"images/3602_q0.png"}],"answer":"$$\\frac{3 \\sqrt{7}}{7}$$"}
{"id":"3611","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$A B = B C = 1$$, points $$D$$ and $$E$$ are moving points on sides $$A B$$ and $$B C$$ respectively, satisfying $$A D = B E$$. Connect $$A E$$ and $$C D$$, then the minimum value of $$A E + C D$$ is         ."},{"type":"image_path","image_path":"images/3611_q0.png"}],"answer":"$$\\sqrt{5}$$"}
{"id":"3649","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$\\triangle BPC$$ is an equilateral triangle, and the extensions of $$BP$$ and $$CP$$ intersect $$AD$$ at points $$E$$ and $$F$$, respectively. Connect $$BD$$ and $$DP$$; $$BD$$ intersects $$CF$$ at point $$H$$. The following conclusions are given: ① $$AE = \\frac{1}{2} FC$$; ② $$\\angle PDE = 15^\\circ$$; ③ $$\\frac{S_{\\triangle DHC}}{S_{\\triangle BHC}} = \\frac{\\sqrt{3}}{3}$$; ④ $$DE^2 = PF \\cdot FC$$. Among these, the correct ones are      (fill in the serial numbers)"},{"type":"image_path","image_path":"images/3649_q0.png"}],"answer":"①②③④"}
{"id":"3654","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$ with side length $$4 \\sqrt{3}$$, moving points D and E lie on sides $$BC$$ and $$AC$$ respectively, such that $$AE = CD$$. Connect $$BE$$ and $$AD$$, intersecting at point P. Then the minimum value of $$CP$$ is          ."},{"type":"image_path","image_path":"images/3654_q0.png"}],"answer":"4"}
{"id":"3657","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$ABCD$$, $$E$$ is the midpoint of $$BC$$. After folding $$\\triangle ABE$$ to obtain $$\\triangle AFE$$, point $$F$$ lies inside the rectangle. Extend $$AF$$ to intersect $$CD$$ at point $$H$$. If $$AD = 4$$ and $$CH = \\frac{4}{3}$$, then the length of the fold line $$AE$$ is      ."},{"type":"image_path","image_path":"images/3657_q0.png"}],"answer":"$$\\sqrt{13}$$"}
{"id":"3672","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square ABCD is 4, and point E is any point on diagonal BD (not coinciding with B or D). Connect AE, and from point E draw EF ⊥ AE, intersecting segment BC at point F. Construct rectangle AEFG with AE and EF as adjacent sides. Connect BG. The following four conclusions are given:\n① AE = EF; ② CD - BF = (√2)/2 * BE;\n③ Let the perimeter of quadrilateral AGBE be m, then 8√2 ≤ m < 8 + 4√2;\n④ When BF = 1, the area of triangle AGB is 3.\nAmong these, the correct conclusions are            . (Fill in all the serial numbers of the correct conclusions)"},{"type":"image_path","image_path":"images/3672_q0.png"}],"answer":"①③④"}
{"id":"3685","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, O is the intersection point of the diagonals, M and N are moving points on sides $$A D$$ and $$C D$$ respectively, and $$A M = C N$$. Connect $$O M$$ and $$B N$$; then the minimum value of $$O M + B N$$ is                  ."},{"type":"image_path","image_path":"images/3685_q0.png"}],"answer":"$$2 \\sqrt{10}$$"}
{"id":"3688","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle B = 60^\\circ$$, point E and point F are moving points on segments $$A B$$ and $$A D$$ respectively, and $$B E = A F$$. Let the side length of rhombus $$A B C D$$ be $$2 a$$, and the perimeter of $$\\triangle A E F$$ be $$m$$. Then the range of the perimeter $$m$$ of $$\\triangle A E F$$ is      . (Expressed in terms of $$a$$)"},{"type":"image_path","image_path":"images/3688_q0.png"}],"answer":"$$2 a + \\sqrt{3} a \\leq m < 4 a$$"}
{"id":"3699","difficulty":"0.4","question_list":[{"type":"text","text":"Given that quadrilateral $$A B C D$$ is a parallelogram, $$A B = 15$$, $$A D = 20$$, $$A H \\bot B C$$ at point H, $$A H = 12$$, point E is a point on segment $$C D$$, connect $$A E$$, fold $$\\triangle A D E$$ along $$A E$$ to obtain $$\\triangle A F E$$, point D lands at point F on the extension of $$B C$$, $$A F$$ intersects $$C D$$ at point G, then $$S_{\\triangle A G E} =$$      ."},{"type":"image_path","image_path":"images/3699_q0.png"}],"answer":"$$\\frac{128}{3}$$"}
{"id":"3703","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, points E and F are on sides BC and CD of rectangle ABCD, respectively. Triangle CEF is folded along line EF to obtain triangle GEF, and then triangle BEG is folded along line BG; the image of point E, point H, falls exactly on diagonal BD. If at this time points F, G, and H are collinear, and segment HF is also symmetric to segment HD with respect to some line, then the value of $$\\frac{BD}{EF}$$ is      ."},{"type":"image_path","image_path":"images/3703_q0.png"}],"answer":"$$2 + \\sqrt{3}$$"}
{"id":"3720","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$ABC$$, $$\\angle ABC = 90^\\circ$$, $$AB = 6$$, segment $$PQ$$ moves along the hypotenuse $$AC$$, and $$PQ = 2$$. Connect $$BP$$ and $$BQ$$. What is the minimum perimeter of $$\\triangle BPQ$$?"},{"type":"image_path","image_path":"images/3720_q0.png"}],"answer":"$$2 \\sqrt{19} + 2$$/$$2 + 2 \\sqrt{19}$$"}
{"id":"3729","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle$$ABC, AD and CE are medians. If the area of quadrilateral BDFE is 6, then the area of $$\\triangle$$ABC is         ."},{"type":"image_path","image_path":"images/3729_q0.png"}],"answer":"18"}
{"id":"3735","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in isosceles triangle ABC, AB = AC = 5, BC = 8. E is a moving point on BC. Triangle ABE is folded along AE to triangle ADE. Then side AC is folded to coincide with AD, with crease AF. When triangle DEF is isosceles, the length of BE is ______."},{"type":"image_path","image_path":"images/3735_q0.png"}],"answer":"$$\\frac{5}{2}$$ or $$\\frac{25}{8}$$ or $$\\frac{7}{4}$$."}
{"id":"3738","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, $$\\angle ABC = 135^\\circ, \\angle BCD = 120^\\circ, CD = 2\\sqrt{3}, AB = \\sqrt{2}, BC = 3 - \\sqrt{3}$$, then the perimeter of quadrilateral $$ABCD$$ is           ."},{"type":"image_path","image_path":"images/3738_q0.png"}],"answer":"$$3 + \\sqrt{2} + \\sqrt{3} + 2 \\sqrt{5}$$"}
{"id":"3743","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 2$$, diagonals $$AC$$ and $$BD$$ intersect at point O. Points E and F are moving points on sides $$BC$$ and $$CD$$ respectively (points E and F do not coincide with endpoints of segments $$BC$$ and $$CD$$) and satisfy $$BE = CF$$. Connect $$OE$$, $$OF$$, and $$EF$$. As points E and F move, the following four conclusions are given: ① $$\\triangle OEF$$ is an isosceles right triangle; ② The minimum area of $$\\triangle OEF$$ is $$\\frac{1}{2}$$; ③ There exists at least one $$\\triangle ECF$$ such that the perimeter of $$\\triangle ECF$$ is $$2 + \\sqrt{3}$$; ④ The area of quadrilateral $$OECF$$ is 1. Write the serial numbers of the correct conclusions        ."},{"type":"image_path","image_path":"images/3743_q0.png"}],"answer":"①②③④"}
{"id":"3799","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 8$$, points $$E, F$$ are two moving points on sides $$A B$$ and $$B C$$ respectively, and $$E F = 2$$, point $$G$$ is the midpoint of $$E F$$, point $$H$$ is a moving point on side $$A D$$, connect $$C H$$ and $$G H$$, then the minimum value of $$G H + C H$$ is           ."},{"type":"image_path","image_path":"images/3799_q0.png"}],"answer":"$$2 \\sqrt{41} - 1$$"}
{"id":"3818","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A = 30^\\circ$$, $$\\angle A C B = 90^\\circ$$, $$B C = 2$$, D is a moving point on $$A B$$, rotate segment $$C D$$ counterclockwise by 90° about point C to obtain segment $$C E$$, connect $$B E$$, then the minimum value of $$B E$$ is     ."},{"type":"image_path","image_path":"images/3818_q0.png"}],"answer":"$$\\sqrt{3} - 1$$/$$- 1 + \\sqrt{3}$$"}
{"id":"3824","difficulty":"0.4","question_list":[{"type":"text","text":"In parallelogram $$A B C D$$, $$\\angle A B C$$ is an acute angle. Fold $$C D$$ along line $$l$$ onto the line containing $$A B$$, with corresponding points $$C^{'}$$ and $$D^{'}$$. If $$A C^{'} : A B : B C = 1 : 3 : 7$$, and from point F draw a perpendicular to $$A B$$ intersecting it at E, then $$\\frac{B E}{B F} =$$         ."},{"type":"image_path","image_path":"images/3824_q0.png"}],"answer":"$$\\frac{2}{7}$$ or $$\\frac{4}{7}$$/$$\\frac{4}{7}$$ or $$\\frac{2}{7}$$"}
{"id":"3826","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AC = 2 + 2$$\\sqrt{3}$$, ∠BAC = 45°, ∠ACB = 30°. Rotate △ABC counterclockwise about point B to obtain $$\\triangle A_{1} B C_{1}$$, where point E is the midpoint of segment AB, and point P is a moving point on segment AC. During the counterclockwise rotation of △ABC about point B, the corresponding point of P is point $$P_{1}$$. The maximum value of segment $$E P_{1}$$ is     , and the minimum value is        ."},{"type":"image_path","image_path":"images/3826_q0.png"}],"answer":"$$4 + \\sqrt{2}$$/$$\\sqrt{2} + 4$$     $$2 - \\sqrt{2}$$/$$- \\sqrt{2} + 2$$"}
{"id":"3829","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle C = 90 \\circ$$, $$A B = B C = 10 \\sqrt{3}$$, $$A D = C D = 10$$, points $$E$$ and $$F$$ lie on sides $$A D$$ and $$B C$$ respectively, connect $$E F$$, point $$G$$ is the midpoint of $$E F$$, connect $$A G$$, if $$A E = 6$$, then the minimum value of $$A G$$ is      ."},{"type":"image_path","image_path":"images/3829_q0.png"}],"answer":"9"}
{"id":"3831","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$ with side length 2, points $$E$$ and $$F$$ lie on sides $$AB$$ and $$BC$$ respectively, $$AF \\bot DE$$, with foot of perpendicular at point $$G$$. Construct rectangle $$DG FH$$ with sides $$DG$$ and $$GF$$. If the area of the shaded region in the figure is 3, then the area of rectangle $$DG FH$$ is     ."},{"type":"image_path","image_path":"images/3831_q0.png"}],"answer":"3"}
{"id":"3840","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A B = B C = \\frac{5 \\sqrt{2}}{2}$$, $$C D = 3$$, $$\\angle A B C = \\angle A D C = 90 \\circ$$, $$E$$ is a point on $$B D$$ such that $$\\angle A E C = 135 \\circ$$, then $$A E =$$      ."},{"type":"image_path","image_path":"images/3840_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"3847","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A B = 2 \\sqrt{3}$$, $$\\angle A B C = 60 \\circ$$, point E is a moving point on diagonal $$B D$$ (not coinciding with point B), and $$B E < \\frac{1}{2} B D$$, connect $$C E$$ intersecting the extension of $$D A$$ at point F."},{"type":"image_path","image_path":"images/3847_q0.png"},{"type":"text","text":"①$$\\angle A F E = \\angle B A E$$;② When $$\\triangle A E F$$ is a right triangle, $$B E = 2$$;③ When $$\\triangle A E F$$ is an isosceles triangle, $$\\angle A F C = 20 \\circ$$ or $$\\angle A F C = 40 \\circ$$;④ Connect $$B F$$, when $$B E = C E$$, $$F C$$ bisects $$\\angle A F B$$.Which of the above conclusions are correct? (Fill in the correct serial numbers.)"}],"answer":"①②③④"}
{"id":"3853","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle B = 60 \\circ$$, $$A B = 4$$, $$E$$ and $$F$$ are points on sides $$A B$$ and $$B C$$ respectively. Fold $$\\triangle E B F$$ along $$E F$$ so that the image of point $$B$$, denoted $$B'$$, lies on side $$A D$$. If $$A E = A B'$$, then the length of $$C F$$ is      ."},{"type":"image_path","image_path":"images/3853_q0.png"}],"answer":"$$4 - 2 \\sqrt{3}$$/$$- 2 \\sqrt{3} + 4$$"}
{"id":"3868","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram. Extend $$A D$$ to point E such that $$D E = A D$$, and $$B E \\bot D C$$. If $$\\triangle A D B$$ is an equilateral triangle with side length 3, and points P, M, N move along segments BE, BC, and CE respectively, then the minimum value of $$P M + P N$$ is      ."},{"type":"image_path","image_path":"images/3868_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{2}$$"}
{"id":"3873","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A D \\bot C D, B C \\bot C D$$, point $$E$$ is the midpoint of $$A B$$, $$B C = C D = 6, A B = 2 \\sqrt{10}$$, point $$F$$ lies on side $$C D$$, and the area of quadrilateral $$A E F D$$ is 9. Find the value of $$\\frac{D F}{F C}$$."},{"type":"image_path","image_path":"images/3873_q0.png"}],"answer":"$$\\frac{1}{4}$$/$$0 . 25$$"}
{"id":"3878","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle$$ABC, $$\\angle B A C = 60 \\circ$$, $$\\angle A B C = 45 \\circ$$, AD bisects $$\\angle B A C$$ and intersects BC at point D, P is a moving point on line AB. Connect DP, and construct parallelogram DPQB with DP and DB as adjacent sides. Connect CQ. If $$A C = 6$$, then the minimum value of CQ is      ."},{"type":"image_path","image_path":"images/3878_q0.png"}],"answer":"$$3 + 3 \\sqrt{3}$$"}
{"id":"3893","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$A C = B C, \\angle A C B = 90 \\circ$$, if $$\\angle C A E = 2 \\angle A B D, A E = 5, E C = 3$$, then the length of $$C D$$ is       ."},{"type":"image_path","image_path":"images/3893_q0.png"}],"answer":"2"}
{"id":"3895","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given equilateral triangle $$\\triangle ABC$$ with side length 6, point D lies on the extension of $$BC$$ such that $$CD = 4$$, and point E is a moving point on line $$AC$$. Connect $$DE$$. Let F and G be the midpoints of $$AB$$ and $$DE$$, respectively. Connect $$FG$$. What is the minimum length of segment $$FG$$?"},{"type":"image_path","image_path":"images/3895_q0.png"}],"answer":"$$\\frac{5}{2} \\sqrt{3}$$"}
{"id":"3896","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A D$$ is an angle bisector of $$\\triangle A B C$$, $$E$$ is the midpoint of $$A D$$, and $$B E$$ is connected. If $$B E = B C$$ and $$C D = 2$$, then $$B D =$$      ."},{"type":"image_path","image_path":"images/3896_q0.png"}],"answer":"$$\\frac{\\sqrt{17} + 1}{2}$$"}
{"id":"3907","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle B C D = 90 \\circ$$, diagonals $$A C, B D$$ intersect at point $$O$$. If $$A B = A C = 5, B C = 6, \\angle A D B = 2 \\angle C B D$$, then the length of $$A D$$ is          ."},{"type":"image_path","image_path":"images/3907_q0.png"}],"answer":"$$\\frac{\\sqrt{97}}{3}$$/$$\\frac{1}{3} \\sqrt{97}$$"}
{"id":"3913","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is inscribed in $$\\bigodot O$$, diameter $$A C$$ intersects chord $$B D$$ at point E, and the extension of $$B D$$ intersects the tangent at point C at point F. Connect $$C D$$. If $$B D = \\frac{8}{3} D E$$, $$C F = 3$$, $$D F = 1$$, then $$B F =$$     , $$A B =$$        ."},{"type":"image_path","image_path":"images/3913_q0.png"}],"answer":"9     $$\\frac{5}{14} \\sqrt{154}$$/$$\\frac{5 \\sqrt{154}}{14}$$"}
{"id":"3924","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, a large equilateral triangle $$ABC$$ is formed by three congruent triangles ($$\\triangle ABE$$, $$\\triangle BCF$$, $$\\triangle CAD$$) and a small equilateral triangle $$DEF$$ in the center. Connect $$BD$$ and extend it to intersect $$AC$$ at point $$G$$. If $$AE = ED = 2$$, then:\n(1) The measure of $$\\angle FDB$$ is      ;\n(2) The length of $$DG$$ is      ."},{"type":"image_path","image_path":"images/3924_q0.png"}],"answer":"$$30 \\circ$$     $$\\frac{4 \\sqrt{3}}{5}$$"}
{"id":"3936","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point $$D$$ is a point on side $$A C$$, with $$C D : A D = 1 : 2$$. Connect $$B D$$, point $$E$$ is a point on segment $$B D$$, with $$B E : E D = 1 : 3$$. Connect $$A E$$, point $$F$$ is the midpoint of segment $$A E$$. Connect $$C F$$, intersecting segment $$B D$$ at point $$G$$. If the area of $$\\triangle A B C$$ is 12, then the area of $$\\triangle E F G$$ is        ."},{"type":"image_path","image_path":"images/3936_q0.png"}],"answer":"$$\\frac{9}{4}$$"}
{"id":"3958","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, construct $$\\triangle A C D$$ with side $$A C$$ such that $$A D = A C$$, point E is a point on $$B C$$, connect AE, $$\\angle B A E = \\frac{1}{2} \\angle C A D$$, connect $$D E$$ . Which of the following conclusions is correct? (Fill in the serial number)\n① $$A C \\bot D E$$; ② $$\\angle A D E = \\angle A C B$$; ③ If $$C D / / A B$$, then $$A E \\bot A D$$; ④ $$D E = C E + 2 B E$$."},{"type":"image_path","image_path":"images/3958_q0.png"}],"answer":"②③④"}
{"id":"3968","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the circle $$\\bigodot O$$ with diameter $$AB$$ is perpendicular to chord $$CD$$ at point $$E$$. From point $$D$$, draw $$DF \\bot AC$$ intersecting at point $$F$$, intersecting $$\\bigodot O$$ at point $$G$$, and intersecting $$AB$$ at point $$H$$. Connect $$AG$$, $$CG$$, and $$EF$$. Given $$CG = \\sqrt{5}$$ and $$AH = 3$$, then $$AG =$$     , and the length of segment $$EF =$$        ."},{"type":"image_path","image_path":"images/3968_q0.png"}],"answer":"3     2"}
{"id":"3978","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle B A C = 90 \\circ, A B = 3, B C = 5$$, point P is any point on side $$B C$$, connect $$P A$$, construct parallelogram $$P A Q C$$ with $$P A$$ and $$P C$$ as adjacent sides, connect $$P Q$$, then the minimum length of $$P Q$$ is         ."},{"type":"image_path","image_path":"images/3978_q0.png"}],"answer":"$$\\frac{12}{5}$$/2.4"}
{"id":"3984","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = 5, AC = 8, BO and CO bisect ∠ABC and ∠ACB respectively. Draw DE ∥ BC through point O. Then the perimeter of △ADE is                ."},{"type":"image_path","image_path":"images/3984_q0.png"}],"answer":"13"}
{"id":"3985","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, points A and B are two points on $$\\bigodot O$$, connect $$AB$$, diameter $$CD$$ is perpendicular to $$AB$$ at point E, point F is on $$\\bigodot O$$, connect $$AF$$, $$BF$$, draw a perpendicular from point A to $$BF$$, intersecting $$BF$$ at point G and intersecting $$\\bigodot O$$ at point H. If $$AE = 3$$, $$CD = 4\\sqrt{3}$$, $$GH = \\sqrt{2}$$, then the length of $$OE$$ is       , and the length of $$AF$$ is          ."},{"type":"image_path","image_path":"images/3985_q0.png"}],"answer":"$$\\sqrt{3}$$     $$2 \\sqrt{10}$$"}
{"id":"3994","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 4, BC = 3, E is a moving point on side AB, and a square DEFG is constructed to the right with DE as a side. Connect CF; then the minimum value of CF is      ."},{"type":"image_path","image_path":"images/3994_q0.png"}],"answer":"$$5 \\sqrt{2}$$"}
{"id":"4001","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the square $$A B C D$$ has side length 4, $$L$$ is the midpoint of $$C D$$, and $$Y, B, D$$ lie on $$\\bigodot C$$, the maximum value of $$\\left|\\sqrt{2} L Y - Y A\\right|$$ is, and the minimum value of $$2 \\sqrt{2} L Y + Y A$$ is"},{"type":"image_path","image_path":"images/4001_q0.png"}],"answer":"$$2 \\sqrt{2}$$     $$4 \\sqrt{5}$$"}
{"id":"4007","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A B = B C, \\angle A B C = \\angle C D A = 90^{\\circ}, B E \\bot A D$$ at $$E, S_{A B C D} = 10$$, then the length of $$B E$$ is"},{"type":"image_path","image_path":"images/4007_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"4014","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C, \\triangle C D E$$ are both equilateral triangles. Rotate $$\\triangle C D E$$ about point $$C$$ such that points $$A, D, E$$ lie on the same straight line. Connect $$B E$$. If $$B E = 1, A E = 4$$, then the length of $$C E$$ is      ."},{"type":"image_path","image_path":"images/4014_q0.png"}],"answer":"3"}
{"id":"4015","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point $$E$$ is the midpoint of $$A D$$, and point $$F$$ is a point on $$A B$$. After folding $$\\triangle A E F$$ along $$E F$$, point $$A$$ lands exactly at point $$G$$ on $$C F$$. Draw $$F H \\parallel A D$$ intersecting $$E G$$ at point $$H$$. If $$A B = 16$$ and $$A D = 24$$, then $$G H =$$      ."},{"type":"image_path","image_path":"images/4015_q0.png"}],"answer":"$$\\frac{21}{8}$$/$$2 \\frac{5}{8}$$/2.625"}
{"id":"4053","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point C is a moving point outside the straight line AB, with AB = 6. Connect CA and CB. Points D and E are the midpoints of AB and BC, respectively. Connect AE and CD, intersecting at point F. When the area of quadrilateral BEFD is 5, the minimum length of segment AC is      ."},{"type":"image_path","image_path":"images/4053_q0.png"}],"answer":"5"}
{"id":"4067","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, $$AD$$ and $$BC$$ are tangent to $$\\bigodot O$$ at points $$A$$ and $$B$$ respectively, $$CD$$ passes through a point $$E$$ on $$\\bigodot O$$, $$AD = DE$$, if $$AB = 12$$, $$BC = 4$$, then the length of $$AD$$ is      ."},{"type":"image_path","image_path":"images/4067_q0.png"}],"answer":"$$9$$"}
{"id":"4091","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$ABCD$$, $$AB = 6$$, $$AD = 4$$, point $$O$$ is a point on side $$AB$$ such that $$AO = 2$$, and point $$E$$ is a moving point on side $$BC$$. Rotate segment $$OE$$ clockwise around point $$O$$ by $$120^{\\circ}$$ to obtain segment $$OE'$$, which intersects side $$AD$$ at point $$F$$. Connect $$EF$$."},{"type":"image_path","image_path":"images/4091_q0.png"},{"type":"text","text":"(1) When point $$E$$ coincides with point $$B$$, the area of $$\\triangle EOF$$ is          . (2) When point $$E$$ moves along side $$BC$$, the minimum area of $$\\triangle EOF$$ is          ."}],"answer":"$$4 \\sqrt{3}$$     $$\\frac{8 \\sqrt{3}}{3}$$"}
{"id":"4098","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$A B = 6$$, $$\\angle A C B = 30 \\circ$$, $$E$$ is the midpoint of $$B C$$, and $$\\triangle A B C$$ is folded along side $$A C$$ to obtain $$\\triangle A F C$$. Points $$M$$ and $$N$$ are two moving points on side $$A C$$ such that $$M N = 2$$. The minimum perimeter of quadrilateral $$B E N M$$ is        ."},{"type":"image_path","image_path":"images/4098_q0.png"}],"answer":"$$2 + 3 \\sqrt{3} + \\sqrt{67}$$"}
{"id":"4105","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the given $$\\bigodot O$$, the perpendicular distance from the center to chord $$AB$$ is $$OH = 6$$, $$CD = 16$$, and point $$E$$ lies on chord $$CD$$ such that $$OE = ED = 5$$. The maximum area of $$\\triangle EAB$$ is  , and at this time, the length of $$DH$$ is     ."},{"type":"image_path","image_path":"images/4105_q0.png"}],"answer":"$$22 \\sqrt{11}$$     $$2 \\sqrt{53}$$"}
{"id":"4107","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C > 90 \\circ$$, $$A B = \\sqrt{13}$$, $$B C = 10$$, point D is the midpoint of $$B C$$, point E lies on $$A C$$, and $$\\triangle C D E$$ is folded along $$D E$$ such that point C lands exactly at point F on the extension of $$B A$$. Connect $$A D$$, given that $$S_{\\Delta E D F} = \\frac{5}{2}$$, find the length of $$A D$$."},{"type":"image_path","image_path":"images/4107_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"4122","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A = m \\circ$$, the angle bisectors of $$\\angle A B C$$ and $$\\angle A C D$$ intersect at point $$A_{1}$$, yielding $$\\angle A_{1}$$; the angle bisectors of $$\\angle A_{1} B C$$ and $$\\angle A_{1} C D$$ intersect at point $$A_{2}$$, yielding $$\\angle A_{2}$$; .. the angle bisectors of $$\\angle A_{2020} B C$$ and $$\\angle A_{2020} C D$$ intersect at point $$A_{2021}$$, yielding $$\\angle A_{2021}$$; the angle bisectors of $$\\angle A_{2021} B C$$ and $$\\angle A_{2021} C D$$ intersect at point $$A_{2022}$$, yielding $$\\angle A_{2022}$$. Then $$\\angle A_{2022} =$$        degrees."},{"type":"image_path","image_path":"images/4122_q0.png"}],"answer":"$$\\frac{m}{2^{2022}}$$"}
{"id":"4154","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90^\\circ$$, $$AC = 3$$, $$BC = 4$$, there is a point P on segment $$BC$$ (not coinciding with B or C). Connect $$AP$$, and fold $$\\triangle ACP$$ along $$AP$$ to obtain $$\\triangle ADP$$. Connect $$CD$$ and $$BD$$. When $$\\triangle CDB$$ is an isosceles triangle, the length of $$CP$$ is        ."},{"type":"image_path","image_path":"images/4154_q0.png"}],"answer":"$$\\frac{6}{5} \\sqrt{5}$$ or $$\\frac{9 - 3 \\sqrt{5}}{2}$$"}
{"id":"4196","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the square $$ABCD$$ with side length 2, points E and F are moving points on sides $$AD$$ and $$CD$$ respectively (not coinciding with the endpoints). Connect $$BE$$ and $$BF$$, intersecting the diagonal $$AC$$ at points P and Q respectively. As points E and F move, $$\\angle EBF = 45^\\circ$$ always holds. Connect $$EF$$, $$PF$$, and $$PD$$. The following conclusions: ① $$PB = PD$$; ② $$\\angle EFD = 2\\angle FBC$$; ③ $$PQ = PA + CQ$$; ④ $$\\triangle BPF$$ is an isosceles right triangle; ⑤ If a perpendicular $$BH \\bot EF$$ is drawn from point B with foot at H, and $$DH$$ is connected, then the minimum value of $$DH$$ is $$2\\sqrt{2} - 2$$. The sequence numbers of all correct conclusions are    ."},{"type":"image_path","image_path":"images/4196_q0.png"}],"answer":"①②④⑤"}
{"id":"4199","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, chord $$CD \\bot AB$$ at point $$E$$, point $$F$$ lies on the circle, and $$\\overset{⌢}{DF} = \\overset{⌢}{CD}$$, $$BE = 2$$, $$CD = 8$$, $$CF$$ intersects $$AB$$ at point $$G$$. Then the length of chord $$CF$$ is                  , and the length of $$AF$$ is                     ."},{"type":"image_path","image_path":"images/4199_q0.png"}],"answer":"$$\\frac{48}{5}$$     $$\\frac{4 \\sqrt{5}}{5}$$/$$\\frac{4}{5} \\sqrt{5}$$"}
{"id":"4223","difficulty":"0.4","question_list":[{"type":"text","text":"Given $$\\triangle A B C$$, $$A B = A C$$, $$A D \\bot B C$$, point F lies on $$A C$$, construct $$E F \\bot A B$$ at E, intersecting the extension of $$B C$$ at G, connect $$E D$$, $$\\angle G F C = 2 \\angle E D A$$, $$D H = C G = 2$$, then the length of $$A F$$ is      ."},{"type":"image_path","image_path":"images/4223_q0.png"}],"answer":"$$\\frac{4}{3} \\sqrt{5}$$/$$\\frac{4 \\sqrt{5}}{3}$$"}
{"id":"4224","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$AB = 5$$, the radius of $$\\bigodot B$$ is 2, point C lies on $$\\bigodot B$$, and segment $$AC$$ is connected. Rotate segment $$AC$$ clockwise around point A by $$60^\\circ$$ to obtain segment $$AD$$. When point C moves along $$\\bigodot B$$, the maximum distance from point D to line $$AB$$ is        ."},{"type":"image_path","image_path":"images/4224_q0.png"}],"answer":"$$\\frac{5}{2} \\sqrt{3} + 2$$"}
{"id":"4280","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = A C$$, rotate $$A C$$ clockwise around point C by $$60 \\circ$$ to obtain $$C D$$, connect BD intersecting $$A C$$ at E, then $$\\frac{A E}{E D} =$$      ."},{"type":"image_path","image_path":"images/4280_q0.png"}],"answer":"$$\\frac{3 \\sqrt{2} - \\sqrt{6}}{6}$$"}
{"id":"4292","difficulty":"0.4","question_list":[{"type":"text","text":"In rectangle $$A B C D$$, $$A B = \\sqrt{3}$$, $$B C = 1$$, fold $$\\triangle A B C$$ along $$A C$$ to obtain $$\\triangle A E C$$, F is a point on $$D C$$, connect $$E F$$, if $$s i n \\angle A E F = \\frac{\\sqrt{21}}{7}$$, then the length of segment $$E F$$ is        ."},{"type":"image_path","image_path":"images/4292_q0.png"}],"answer":"$$\\frac{\\sqrt{21}}{9}$$"}
{"id":"4303","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, $$C$$ is a point on the circle, and $$\\angle AOC = 120^\\circ$$. The radius of $$\\bigodot O$$ is $$4$$. Point $$P$$ is a moving point on the circle, and $$Q$$ is the midpoint of $$AP$$. What is the maximum length of $$CQ$$?"},{"type":"image_path","image_path":"images/4303_q0.png"}],"answer":"$$2 + 2 \\sqrt{7}$$/$$2 \\sqrt{7} + 2$$"}
{"id":"4306","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, extend $$B C$$ to point E such that $$B E = 8$$. A moving point P starts from point B and moves toward point A along the path $$B C - C D - D A$$ at a speed of 2 units per second. Let the motion time of point P be $$t$$ seconds. When $$\\triangle A B P$$ and $$\\triangle D C E$$ are congruent, the value of $$t$$ is          ."},{"type":"image_path","image_path":"images/4306_q0.png"}],"answer":"1 or 7"}
{"id":"4311","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, points D and E are points on sides $$A B$$ and $$A C$$ respectively. Fold $$A D$$ and $$B D$$ along $$D E$$ and $$D C$$ to $$A^{'} D$$. Given that $$\\angle A^{'} C A = 36 \\circ$$ and $$\\angle B + \\frac{\\angle A^{'} E C}{2} = 90 \\circ$$, then $$\\angle A^{'} D C$$ is           $$\\circ$$."},{"type":"image_path","image_path":"images/4311_q0.png"}],"answer":"54"}
{"id":"4320","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point $$D$$ is a point on side $$A B$$, with $$A D : D B = 3 : 1$$, connect $$C D$$, point $$E$$ is a point on segment $$A C$$, with $$A E : E C = 1 : 2$$, connect $$B E$$, $$C D$$ and $$B E$$ intersect at point $$F$$, if $$A C = 8$$, $$B C = 9$$, then the maximum value of the sum of the areas of $$\\triangle B D F$$ and $$\\triangle C E F$$ is      ."},{"type":"image_path","image_path":"images/4320_q0.png"}],"answer":"$$17$$"}
{"id":"4338","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$\\angle B = 30 \\circ$$, $$B C = 4$$, points P and D are moving points on $$B C$$ and $$A B$$ respectively, then the minimum value of $$A P + D P$$ is          ."},{"type":"image_path","image_path":"images/4338_q0.png"}],"answer":"4"}
{"id":"4342","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ and $$\\triangle C E F$$ are both isosceles right triangles, $$\\angle B A C = \\angle C E F = 90 \\circ$$, and point E lies on side $$A C$$. Rotate $$\\triangle C E F$$ counterclockwise about point C by $$\\alpha \\left(\\right. 0 \\circ < \\alpha < 180 \\circ \\left.\\right)$$; during the rotation, line $$E F$$ intersects line $$A C$$ and line $$B C$$ at points M and N, respectively. If $$\\triangle C M N$$ is an isosceles triangle, then the value of $$\\alpha$$ is      ."},{"type":"image_path","image_path":"images/4342_q0.png"}],"answer":"$$22.5 \\circ$$ or $$45 \\circ$$ or $$112.5 \\circ$$"}
{"id":"4347","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, square $$ABCD$$ is folded along line $$EF$$ such that the image of point $$B$$, denoted $$M$$, lies on side $$AD$$, and point $$C$$ lands at point $$N$$. Segment $$MN$$ intersects $$CD$$ at point $$P$$. The crease intersects sides $$AB$$ and $$CD$$ at points $$E$$ and $$F$$, respectively. Connect $$BM$$. If $$\\frac{DP}{CP} = \\frac{1}{2}$$, then the value of $$\\frac{AE}{BE}$$ is        ."},{"type":"image_path","image_path":"images/4347_q0.png"}],"answer":"$$\\frac{12}{13}$$"}
{"id":"4374","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle B = 30 \\circ$$, moving points $$M$$ and $$N$$ are on $$B C$$ and $$A B$$ respectively, and $$A N = 2 B M > 0$$. Connect $$A M$$ and $$C N$$. If $$A C = 1$$, then the minimum value of $$C N + 2 A M$$ is         ."},{"type":"image_path","image_path":"images/4374_q0.png"}],"answer":"$$\\sqrt{17}$$"}
{"id":"4388","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 5, B C = 9$$, E is a point on side $$A B$$, $$A E = 2$$, F is a moving point on line $$B C$$, rotate line segment $$E F$$ counterclockwise by $$90 \\circ$$ about point E to obtain segment $$E G$$, connect $$C G, D G$$, then the minimum value of $$C G + D G$$ is        ."},{"type":"image_path","image_path":"images/4388_q0.png"}],"answer":"13"}
{"id":"4406","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, $$A B = 2$$, $$A H \\bot B C$$ at point $$H$$, and an equilateral triangle $$\\triangle A H M$$ is constructed to the left using $$A H$$ as a side. Let $$Q$$ be a moving point on segment $$A M$$. Connect $$B Q$$ and $$Q H$$. Then the minimum perimeter of $$\\triangle Q B H$$ is                ."},{"type":"image_path","image_path":"images/4406_q0.png"}],"answer":"$$\\sqrt{7} + 1$$"}
{"id":"4410","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AD = 6, AB = 8. Point E is a moving point on side DC. △AD'E is symmetric to △ADE with respect to line AE. When △CD'E is a right triangle, the length of DE is  ."},{"type":"image_path","image_path":"images/4410_q0.png"}],"answer":"3 or 6"}
{"id":"4411","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$\\triangle ABC$$, $$\\angle ACB = 90^\\circ$$, point D lies on side $$AC$$ (excluding points A and C), connect $$BD$$, and construct an adjacent isosceles right triangle $$\\triangle BDE$$ with $$DB$$ as one leg to the right side, $$\\angle BDE = 90^\\circ$$, and connect $$CE$$. If $$AD = 2$$, $$CE = \\sqrt{5}$$, then the length of segment $$BE$$ is    ."},{"type":"image_path","image_path":"images/4411_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"4426","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$ABC$$, $$AC = BC = 1$$, point P lies on the semicircle with hypotenuse $$AB$$ as its diameter, M is the midpoint of $$PC$$, when point P moves along the semicircle from point A to point B, the length of the path traced by point M is       ."},{"type":"image_path","image_path":"images/4426_q0.png"}],"answer":"$$\\frac{\\sqrt{2}}{4} \\pi$$"}
{"id":"4438","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, G is a point on the diagonal AC of square ABCD. Connect BG, and draw EF perpendicular to BG, intersecting AB at point E and CD at point F. If AG = 2√2 and CF = 2, then the area of triangle EGB is     ."},{"type":"image_path","image_path":"images/4438_q0.png"}],"answer":"5"}
{"id":"4447","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$A D = 10$$, point E lies on $$A D$$ such that $$D E = 2$$. Point G is the midpoint of $$A E$$, and point P is a moving point on side $$B C$$. F is the midpoint of $$E P$$. Find the minimum value of $$G F + E F$$       ."},{"type":"image_path","image_path":"images/4447_q0.png"}],"answer":"5"}
{"id":"4449","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the width of rectangle $$A B C D$$ is $$8$$, and its length is $$12$$. Point $$Q$$ lies on segment $$C D$$, with $$C Q = 5$$. Point $$P$$ lies on segment $$B C$$. When $$\\triangle P Q C$$ is folded along $$P Q$$, point $$C$$ lands exactly at point $$R$$ on side $$A D$$. Point $$O$$ lies on segment $$A B$$. When $$\\triangle A O R$$ is folded along $$O R$$, point $$A$$ lands exactly at point $$H$$ on segment $$P R$$. Find the distance from point $$H$$ to segment $$D C$$."},{"type":"image_path","image_path":"images/4449_q0.png"}],"answer":"$$\\frac{44}{5}$$"}
{"id":"4454","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points $$M$$, $$N$$ lie on sides $$C D$$, $$B C$$ respectively, and $$B N = 2 D M$$. Connect $$A M$$, draw $$N P \\bot A M$$ through point $$N$$, with foot at $$P$$, and connect $$D P$$. What is the minimum length of $$D P$$?"},{"type":"image_path","image_path":"images/4454_q0.png"}],"answer":"$$2$$"}
{"id":"4458","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A E$$ is the altitude from side $$B C$$, and points $$F$$ and $$G$$ are moving points on the altitude $$A E$$ and side $$C D$$, respectively, such that $$A F = D G$$. If $$A B = 5$$, $$B C = 4$$, $$\\angle A D C = 60^\\circ$$, then the minimum value of $$B F + A G$$ is      ."},{"type":"image_path","image_path":"images/4458_q0.png"}],"answer":"$$\\sqrt{41}$$"}
{"id":"4487","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B =\\text{60} \\circ$$, $$B C = 12$$. Point $$M$$ lies on side $$B C$$, and $$M C = \\frac{1}{4} B C$$, ray $$C D \\bot B C$$ at point $$C$$, point $$P$$ is a moving point on ray $$C D$$, and point $$N$$ is a moving point on segment $$A B$$."},{"type":"image_path","image_path":"images/4487_q0.png"},{"type":"text","text":"(1) Does the length of segment $$M P + N P$$ have a minimum value?          . (Fill in \"yes\" or \"no\") (2) If the length of segment $$M P + N P$$ has a minimum value, directly write the length of $$B N$$; if not, provide the reason          ."}],"answer":"yes     $$\\frac{15}{2}$$"}
{"id":"4492","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A B = 4$$, $$A D = 6$$, $$\\angle A = 120 \\circ$$, points $$F$$ and $$N$$ are the midpoints of $$C D$$ and $$A B$$, respectively. Point $$E$$ moves along side $$A D$$. Fold $$\\triangle E D F$$ along $$E F$$ such that point $$D$$ lands at $$D^{'}$$. Connect $$B D^{'}$$, and let point $$M$$ be the midpoint of $$B D^{'}$$. Then the minimum value of $$M N$$ is         ."},{"type":"image_path","image_path":"images/4492_q0.png"}],"answer":"$$\\sqrt{7} - 1$$/$$- 1 + \\sqrt{7}$$"}
{"id":"4505","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$\\triangle ABC$$, $$AB = AC = 20$$, $$BC = 32$$, $$\\triangle ABD$$ is an equilateral triangle, and P is a moving point on the angle bisector of $$\\angle BAC$$. Connect $$PC$$ and $$PD$$. The minimum value of $$PC + PD$$ is       ."},{"type":"image_path","image_path":"images/4505_q0.png"}],"answer":"20"}
{"id":"4507","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle paper $$ABCD$$, $$AB = 2$$, $$BC = 2\\sqrt{2}$$, point $$E$$ is the midpoint of $$AB$$, and point $$F$$ is a moving point on side $$AD$$. Fold $$\\triangle AEF$$ along the line $$EF$$ to obtain $$\\triangle A'EF$$. Connect $$A'C$$ and $$A'D$$. When $$\\triangle A'DC$$ is an isosceles triangle, the length of $$AF$$ is        ."},{"type":"image_path","image_path":"images/4507_q0.png"}],"answer":"$$\\frac{\\sqrt{2}}{2}$$ or 1 or $$\\sqrt{2}$$"}
{"id":"4512","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, $$A B = A C$$, $$\\angle B A C = 120^\\circ$$, D is a point inside $$\\triangle A B C$$, and an isosceles triangle $$\\triangle D A E$$ is constructed with $$A D$$ as a leg such that $$\\angle D A E = \\angle B A C$$. Connect $$B E$$ and $$C D$$. If $$M$$ and $$N$$ are the midpoints of $$D E$$ and $$B C$$ respectively, and $$M N = 1$$, then the length of $$C D$$ is       ."},{"type":"image_path","image_path":"images/4512_q0.png"}],"answer":"2"}
{"id":"4537","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 3 \\angle B$$, $$C D$$ bisects $$\\angle A C B$$, $$A E \\bot C D$$, with foot at point E. If $$B D = 11$$, $$A D = 5$$, then (1) $$\\frac{B C}{A C} =$$      ; (2) the perimeter of $$\\triangle A B C$$ is      ."},{"type":"image_path","image_path":"images/4537_q0.png"}],"answer":"$$\\frac{11}{5}$$     $$16 + \\frac{32}{3} \\sqrt{5}$$"}
{"id":"4540","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle B A C = 30 \\circ$$, $$A C = 3$$, $$A D = \\frac{1}{4} A B$$, points $$E$$, $$F$$ lie on segment $$A C$$, an equilateral triangle $$\\triangle A E G$$ is constructed outward from $$\\triangle A B C$$ with $$A E$$ as a side, point $$M$$ is the midpoint of $$E G$$, connect $$D M$$, connect $$B F$$, construct an equilateral triangle $$\\triangle B F N$$ on the right side of $$B F$$, connect $$C N$$, connect $$M N$$, then the minimum value of $$C N + M N + D M$$ is          ."},{"type":"image_path","image_path":"images/4540_q0.png"}],"answer":"$$\\frac{9}{2}$$"}
{"id":"4544","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 8 \\text{cm}$$, $$B C = 14 \\text{cm}$$. Point P starts from point A and moves along the path $$A \\rightarrow C \\rightarrow B$$ toward the endpoint B. Point Q starts from point B and moves along the path $$B \\rightarrow C \\rightarrow A$$ toward the endpoint A. Points P and Q begin moving simultaneously at speeds of $$2 \\text{cm} / \\text{s}$$ and $$3 \\text{cm} / \\text{s}$$, respectively, and both must reach their respective endpoints before stopping. Perpendiculars are drawn from P and Q to line $$l$$, meeting it at points E and F, respectively. Let the time of motion be $$t$$ seconds. To make triangle $$P E C$$ congruent to triangle $$Q F C$$, the value of $$t$$ is        ."},{"type":"image_path","image_path":"images/4544_q0.png"}],"answer":"$$\\frac{22}{5}$$ or 6 or 8"}
{"id":"4550","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle A = 60 \\circ$$, $$G$$ is the midpoint of $$A D$$, point $$E$$ lies on the extension of $$B C$$, $$F$$ and $$H$$ are the midpoints of $$C E$$ and $$G E$$ respectively, $$\\angle E H F = \\angle D G E$$, $$C F = \\sqrt{7}$$, then $$A B =$$     ."},{"type":"image_path","image_path":"images/4550_q0.png"}],"answer":"4"}
{"id":"4568","difficulty":"0.4","question_list":[{"type":"text","text":"$$\\triangle A B C$$ is an isosceles right triangle, $$A B = A C = 6$$, $$\\angle B A C = 90 \\circ$$, point $$D$$ moves along side $$B C$$. Construct an isosceles right triangle $$\\triangle A D E$$ with right angle at $$A$$ on the right side of $$A D$$ (as shown in the figure). $$M$$ is the midpoint of $$D E$$, $$N$$ is a trisection point of $$B C$$, $$C N = \\frac{1}{3} B C$$, connect $$M N$$, then the minimum value of segment $$M N$$ is      ."},{"type":"image_path","image_path":"images/4568_q0.png"}],"answer":"1"}
{"id":"4578","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$A B = 3$$, $$B C = 4$$, $$D$$ is the midpoint of side $$A C$$, connect $$B D$$, fold $$\\triangle A B D$$ along $$B D$$ to obtain $$\\triangle E B D$$, connect $$C E$$, then the distance from point $$E$$ to $$B C$$ is      ."},{"type":"image_path","image_path":"images/4578_q0.png"}],"answer":"$$\\frac{21}{25}$$/$$0 . 84$$"}
{"id":"4591","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 4$$, M is a moving point on side $$A B$$ (excluding endpoints). Fold $$\\triangle A D M$$ along the straight line $$D M$$ to obtain $$\\triangle N D M$$; when ray $$C N$$ intersects segment $$A B$$ at point P, connect $$D P$$, then the maximum value of $$D P$$ is      ."},{"type":"image_path","image_path":"images/4591_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"4595","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point $$E$$ lies on side $$A B$$, with $$B E = B C$$; point $$F$$ lies on side $$C D$$; connect $$E F$$, and fold quadrilateral $$B E F C$$ along $$E F$$, such that point $$B$$ lands exactly at point $$G$$ on side $$A D$$, and the image of point $$C$$ is point $$H$$. Given that $$A B = 3 A G = 9$$, find the length of $$E F$$."},{"type":"image_path","image_path":"images/4595_q0.png"}],"answer":"$$\\frac{5 \\sqrt{10}}{3}$$"}
{"id":"4600","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point $$D$$ is the midpoint of side $$AB$$ of $$\\triangle ABC$$. Folding $$\\triangle BCD$$ along the line $$CD$$ makes it coincide with $$\\triangle ECD$$. If $$AB = 4$$, $$CD = 2$$, $$AE = 1$$, then the distance from point $$C$$ to line $$AB$$ is      ."},{"type":"image_path","image_path":"images/4600_q0.png"}],"answer":"$$\\frac{\\sqrt{15}}{2}$$/$$\\frac{1}{2} \\sqrt{15}$$"}
{"id":"4606","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AC$$ is the diameter of circle O, $$AC = 4$$, $$\\angle ACB = 60^\\circ$$, point D is a moving point on chord $$AB$$, then the minimum value of $$OD + \\frac{1}{2} BD$$ is           ."},{"type":"image_path","image_path":"images/4606_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"4616","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle A B C = \\angle A D C = 45^{\\circ}$$, $$\\angle B A D = \\angle B C A = 105^{\\circ}$$, if $$B C = \\sqrt{6} - \\sqrt{2}$$, then the length of $$C D$$ is        ."},{"type":"image_path","image_path":"images/4616_q0.png"}],"answer":"2"}
{"id":"4634","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of semicircle $$O$$, $$M$$ and $$C$$ are trisection points of the semicircle, $$AB = 8$$, $$BD$$ is tangent to semicircle $$O$$ at point $$B$$, point $$P$$ is a moving point on $$\\overset{⌢}{AM}$$ (not coinciding with points $$A$$ or $$M$$), line $$PC$$ intersects $$BD$$ at point $$D$$, $$BE \\bot OC$$ at point $$E$$, and extending $$BE$$ intersects $$PC$$ at point $$F$$. Which of the following conclusions are correct? (Write all correct conclusion numbers)\n① $$PB = PD$$; ② The length of $$\\overset{⌢}{BC}$$ is $$\\frac{4}{3} \\pi$$; ③ $$\\angle DBE = 45^\\circ$$; ④ $$\\triangle BCF \\sim \\triangle PFB$$; ⑤ $$CF \\cdot CP$$ is constant."},{"type":"image_path","image_path":"images/4634_q0.png"}],"answer":"②⑤"}
{"id":"4644","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$AB = 2 \\sqrt{2}$$, and C is a moving point on segment $$AB$$. On the same side of $$AB$$, construct rhombuses $$ACED$$ and $$CBGF$$ with sides $$AC$$ and $$CB$$, respectively. Points C, E, F lie on a straight line, and $$\\angle D = 120^\\circ$$. P and Q are the midpoints of diagonals $$AE$$ and $$BF$$, respectively. When point C moves along segment $$AB$$, the shortest distance between points P and Q is     (answer in radical form)."},{"type":"image_path","image_path":"images/4644_q0.png"}],"answer":"$$\\frac{\\sqrt{6}}{2}$$"}
{"id":"4645","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, inside the square $ABCD$ with side length $10$ (excluding the boundary) there is a point $E$. Connect $CE$. From point $A$, construct $\\angle BAF = \\angle DCE$, and $AF = CE$. Connect $EF$. Rotate segment $EF$ clockwise around point $E$ by $90^\\circ$, and point $F$ happens to fall on point $D$. Find the length of $EC$."},{"type":"image_path","image_path":"images/4645_q0.png"}],"answer":"$$\\text{2} \\sqrt{\\text{5}}$$"}
{"id":"4646","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, ∠MAN = 90°, point C lies on side AM, AC = 4, point B is a moving point on side AN, connect BC, △A′BC is symmetric to △ABC with respect to the line containing BC, points D and E are the midpoints of AC and BC respectively, connect DE and extend it to intersect the line containing A′B at point F, connect A′E. When △A′EF is a right triangle, the length of AB is     ."},{"type":"image_path","image_path":"images/4646_q0.png"}],"answer":"$$4 \\sqrt{3}$$ or 4"}
{"id":"4653","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $ABCD$ with side length $6$, points $E$ and $F$ are moving points on sides $AB$ and $BC$ respectively, satisfying $AE = BF$. $AF$ intersects $DE$ at point $O$. Point $M$ is the midpoint of $DF$, and point $G$ is on side $AB$ such that $AG = 2GB$. Find the minimum value of $OM + \\frac{1}{2}FG$."},{"type":"image_path","image_path":"images/4653_q0.png"}],"answer":"$$5$$"}
{"id":"4663","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, the angle bisector of $$\\angle B A D$$ intersects $$B C$$ at point $$O$$. With $$O$$ as the center and $$O A$$ as the radius, an arc is drawn, which passes exactly through point $$D$$. The area of the shaded region in the figure is      ."},{"type":"image_path","image_path":"images/4663_q0.png"}],"answer":"$$2 \\pi - \\text{4}$$"}
{"id":"4671","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 8, and M and N are moving points on sides $$A B$$ and $$B C$$, respectively. An isosceles right triangle $$\\text{Rt} \\triangle M P N$$ is constructed with $$M N$$ as the hypotenuse (where $$M P = N P$$, $$\\angle M P N = 90^\\circ$$). Point E lies on side $$C D$$ such that $$D E = 3$$. Connect $$P E$$ and $$P C$$. Find the minimum perimeter of $$\\triangle C P E$$      ."},{"type":"image_path","image_path":"images/4671_q0.png"}],"answer":"$$5 + \\sqrt{73}$$/$$\\sqrt{73} + 5$$"}
{"id":"4673","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, and D is a point in the plane. Connect $$A D$$, and rotate $$A D$$ clockwise around point D by $$60^\\circ$$ to obtain segment $$D E$$. Connect $$B D$$ and $$C E$$. When $$\\angle D A C = 30^\\circ$$, $$A B = 2 \\sqrt{3}$$, and $$A D = 4$$, then $$C E =$$          ."},{"type":"image_path","image_path":"images/4673_q0.png"}],"answer":"2 or $$2 \\sqrt{7}$$"}
{"id":"4675","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$ABCD$$ is a square, and point $$P$$ is a moving point on the circle $$\\bigodot O$$ with diameter $$AD$$. Connect $$BP$$, and construct an equilateral triangle $$BPQ$$ with $$BP$$ as a side. Connect $$OQ$$. If $$AB = 2$$, then the maximum value of segment $$OQ$$ is        ."},{"type":"image_path","image_path":"images/4675_q0.png"}],"answer":"$$\\sqrt{5} + 1$$/$$1 + \\sqrt{5}$$"}
{"id":"4699","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 60 \\circ$$, $$B C = 2$$, $$D$$ is the midpoint of $$A B$$, and $$E$$ is a point on $$A C$$. If $$D E$$ bisects the perimeter of $$\\triangle A B C$$, then the length of $$D E$$ is     ."},{"type":"image_path","image_path":"images/4699_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"4722","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$A B = 6$$, $$\\angle D A C = 60 \\circ$$, point $$F$$ lies on segment $$A O$$, moving from point $$A$$ to point $$O$$, connect $$D F$$, and construct an equilateral triangle $$\\triangle D F E$$ with $$D F$$ as a side, with point $$E$$ and point $$A$$ located on opposite sides of $$D F$$."},{"type":"image_path","image_path":"images/4722_q0.png"},{"type":"text","text":"(1) When point $$F$$ moves to point $$O$$, the length of $$C E$$ is      ; (2) As point $$F$$ moves along segment $$A O$$ from point $$A$$ to point $$O$$, the minimum value of $$C E$$ is      ."}],"answer":"$$6$$     $$3 \\sqrt{3}$$"}
{"id":"4750","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = A D = 6$$, point P moves inside $$\\square A B C D$$, connecting $$P A$$, $$P B$$, $$P C$$. If $$\\angle A P B = \\angle A B C = 60^\\circ$$, then the maximum value of $$P A + P C$$ is           ."},{"type":"image_path","image_path":"images/4750_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"4755","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\Delta A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 6$$, $$B C = 8$$, D and E are two moving points on sides $$B C$$ and $$A C$$ respectively, with $$D E = 4$$, P is the midpoint of $$D E$$, connect $$P A$$ and $$P B$$, then the minimum value of $$P A + \\frac{1}{4} P B$$ is   ."},{"type":"image_path","image_path":"images/4755_q0.png"}],"answer":"$$\\frac{\\sqrt{145}}{2}$$"}
{"id":"4761","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$ABCD$$ is $$\\sqrt{2}$$, and points E and G are the midpoints of sides $$AD$$ and $$BC$$, respectively. Connect $$BE$$, and fold $$\\triangle ABE$$ along $$BE$$ into the same plane to obtain $$\\triangle FBE$$. There is a point H on side $$CD$$; connect $$GH$$, and fold $$\\triangle CGH$$ along $$GH$$ into the same plane to obtain $$\\triangle PGH$$. If point P happens to fall on $$BF$$, then the length of the fold line $$GH$$ is       ."},{"type":"image_path","image_path":"images/4761_q0.png"}],"answer":"$$\\frac{5 \\sqrt{2}}{8}$$"}
{"id":"4766","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠ABC = 90°, ∠ACB = 30°, hypotenuse AC = 4, and point P is a moving point inside the triangle. The minimum value of PA + PB + PC is      ."},{"type":"image_path","image_path":"images/4766_q0.png"}],"answer":"$$2 \\sqrt{7}$$"}
{"id":"4776","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$B C = 6$$, point $$E$$ lies on side $$B C$$, $$C E = 2$$. If points $$P$$ and $$Q$$ are two moving points on sides $$C D$$ and $$A B$$ respectively, and segment $$P Q$$ always satisfies being perpendicular to $$A E$$ with foot of perpendicular at $$F$$, then the minimum value of $$A P + Q E$$ is      ."},{"type":"image_path","image_path":"images/4776_q0.png"}],"answer":"$$5 \\sqrt{5}$$"}
{"id":"4784","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = a$$, points E and F lie on diagonal $$BD$$, and $$\\angle ECF = \\angle ABD$$. After rotating $$\\triangle BCE$$ about point C by a certain angle, we obtain $$\\triangle DCG$$, and connect $$FG$$. The following conclusions are given:\n① $$\\angle FCG = \\angle CDG$$; ② $$BE^{2} + DF^{2} = EF^{2}$$; ③ $$FC$$ bisects $$\\angle BFG$$; ④ The area of $$\\triangle CEF$$ equals $$\\frac{1}{4} a^{2}$$;\nAmong these, the correct conclusions are      . (Fill in all the correct conclusion numbers)"},{"type":"image_path","image_path":"images/4784_q0.png"}],"answer":"①②③"}
{"id":"4800","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in a square grid with side length 1, points A, B, C, D, E are all located at the vertices of the small squares. Connect AB and CD, intersecting at P. According to the auxiliary lines added as indicated in the figure, the value of cos∠BPC is        ."},{"type":"image_path","image_path":"images/4800_q0.png"}],"answer":"$$\\frac{\\sqrt{5}}{5}$$/$$\\frac{1}{5} \\sqrt{5}$$"}
{"id":"4822","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ, \\angle A = 60 \\circ, A C = 1$$, rotate $$\\triangle A B C$$ counterclockwise about point C to obtain $$\\triangle A^{'} B^{'} C$$, at which point $$A^{'}$$ lies exactly on side $$A B$$. The distance between point $$B^{'}$$ and point B is"},{"type":"image_path","image_path":"images/4822_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"4835","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$l_{1}, l_{2}, l_{3}, l_{4}$$ are four parallel lines in the same plane, and the distance between any two adjacent parallel lines is $$h$$. The vertices of a square $$A B C D$$ with area 25 lie on these four lines respectively. What is the value of $$h$$?"},{"type":"image_path","image_path":"images/4835_q0.png"}],"answer":"$$\\sqrt{5}$$"}
{"id":"4847","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus paper $$A B C D$$, $$A B = 1$$, $$\\angle B = 60^\\circ$$. The rhombus paper is folded along the crease $$E F$$, such that point $$D$$ lands at the midpoint $$G$$ of $$A B$$. Then the length of $$D E$$ is      ."},{"type":"image_path","image_path":"images/4847_q0.png"}],"answer":"$$\\frac{7}{10}$$/$$0 . 7$$"}
{"id":"4868","difficulty":"0.6","question_list":[{"type":"text","text":"The Huayang Peak is located in the Huayang Peak Park in Hengyang, and it is the foremost among the seventy-two peaks of Mount Heng. Wang Anshi once wrote a couplet: \"Ten thousand miles away, the geese from Hengyang, regularly return here.\" In front of the peak, the Huayang Peak Square has been developed, with a large goose sculpture at its center, serving as the city emblem of Hengyang. A certain extracurricular practice group measured the height of the goose sculpture using a theodolite and a tape measure, obtaining the following data: as shown in the figure, $$A E = 10 \\text{m}$$, $$\\angle B D G = 30^\\circ$$, $$\\angle B F G = 60^\\circ$$. It is known that the height of the theodolite $$D A$$ is $$1.5 \\text{m}$$. Then the height of the goose sculpture $$B C$$ is approximately         $$\\text{m}$$. (Round the result to $$0.1 \\text{m}$$. Reference data: $$\\sqrt{3} \\approx 1.732$$)"},{"type":"image_path","image_path":"images/4868_q0.png"}],"answer":"10.2"}
{"id":"4907","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, BD is a diagonal, AE ⊥ BD, with foot at E, connect CE. If $$\\tan \\angle ADB = \\frac{1}{2}$$, then the value of $$\\tan \\angle DEC$$ is        ."},{"type":"image_path","image_path":"images/4907_q0.png"}],"answer":"$$\\frac{2}{3}$$"}
{"id":"4911","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, D and E are points on $$A B$$ and $$A C$$ respectively. Fold $$\\triangle B C D$$ and $$\\triangle A D E$$ along $$C D$$ and $$D E$$ respectively, such that points A and B coincide exactly at point $$A^{'}$$. Then $$\\angle E A^{'} C =$$        °. If $$B C = 3$$, $$A C = 5$$, then $$A E =$$"},{"type":"image_path","image_path":"images/4911_q0.png"}],"answer":"$$90 \\circ$$     $$\\frac{8}{5}$$"}
{"id":"4918","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in sector $$A O B$$, $$\\angle A O B = 90 \\circ$$, $$O A = 2$$, point $$C$$ is a point on $$O B$$. The sector $$A O B$$ is folded along $$A C$$, such that the image of point $$B$$, denoted $$B'$$, lies on ray $$A O$$. The area of the shaded region in the figure is         ."},{"type":"image_path","image_path":"images/4918_q0.png"}],"answer":"π+4–4$$\\sqrt{2}$$"}
{"id":"4922","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 4 \\sqrt{3}$$, an arc is drawn with center at point B and radius AB, intersecting AC at point E and BC at point F. The area of the shaded region in the figure is   ."},{"type":"image_path","image_path":"images/4922_q0.png"}],"answer":"$$\\frac{4 \\pi}{3}$$/$$\\frac{4}{3} \\pi$$"}
{"id":"4930","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of square OABC is 2. Rotate square OABC counterclockwise about point O by an angle α (0° < α < 180°) to obtain square OA′B′C′. Connect BC′. When point A′ lies exactly on segment BC′, the length of segment BC′ is    ."},{"type":"image_path","image_path":"images/4930_q0.png"}],"answer":"$$\\sqrt{6} + \\sqrt{2}$$"}
{"id":"4938","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, E is the midpoint of side $$A B$$, and connecting $$D E$$ intersects diagonal $$A C$$ at point F. If $$A B = 8$$, $$A D = 6$$, then the length of $$A F$$ is      ."},{"type":"image_path","image_path":"images/4938_q0.png"}],"answer":"$$\\frac{10}{3}$$/$$3 \\frac{1}{3}$$"}
{"id":"4949","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, $$CD$$ is a chord of $$\\bigodot O$$, connect $$AC$$, $$AD$$, if $$\\angle ACD = 65^\\circ$$, then the measure of $$\\angle BAD$$ is       ."},{"type":"image_path","image_path":"images/4949_q0.png"}],"answer":"$$25 \\circ$$"}
{"id":"4953","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point $$E$$ is the midpoint of side $$A B$$. Fold $$\\triangle A D E$$ along $$D E$$ to obtain $$\\triangle F D E$$. Extend $$E F$$ to intersect $$B C$$ at point $$G$$, and connect $$B F$$ and $$D G$$."},{"type":"image_path","image_path":"images/4953_q0.png"},{"type":"text","text":"(1) If $$\\angle E D F = \\alpha$$, then the measure of $$\\angle E B F$$ is      . (Express in terms of $$\\alpha$$) (2) If $$A B : A D = 2 : 3$$, then the tangent of $$\\angle G E B$$ is      ."}],"answer":"$$90 \\circ - \\alpha$$     $$\\frac{3}{4}$$/$$0.75$$"}
{"id":"4955","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the paper of right triangle $$\\text{Rt} \\triangle A B C$$, ∠C = 90°, AC = 7, AB = 25, point D lies on side BC. Triangle △ADB is folded along AD to obtain triangle △ADB′, and AB′ intersects side BC at point E. If $$\\triangle E D B^{'}$$ is a right triangle, then the length of BD is     ."},{"type":"image_path","image_path":"images/4955_q0.png"}],"answer":"17 or $$\\frac{75}{4}$$"}
{"id":"4997","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A D = 5, A B = 12, s i n A = \\frac{4}{5}$$. Draw $$D E \\bot A B$$, with foot at E, then $$s i n \\angle B C E =$$      ."},{"type":"image_path","image_path":"images/4997_q0.png"}],"answer":"$$\\frac{9 \\sqrt{10}}{50}$$"}
{"id":"5011","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C, \\angle B A C = 90 \\circ$$.Line l passes through point A. From point B, draw $$B E \\bot l$$ meeting it at point E; from point C, draw $$C F \\bot l$$ meeting it at point F. If $$B E = 2, C F = 5$$, then $$E F =$$          ."},{"type":"image_path","image_path":"images/5011_q0.png"}],"answer":"7"}
{"id":"5019","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 4$$, M is a moving point on side $$A B$$ (excluding endpoints). Fold $$\\triangle A D M$$ along the straight line $$D M$$ to obtain $$\\triangle N D M$$. When ray $$C N$$ intersects segment $$A B$$ at point P, connect $$D P$$. Then the area of $$\\triangle C D P$$ is           ; the maximum value of $$D P$$ is           ."},{"type":"image_path","image_path":"images/5019_q0.png"}],"answer":"$$10$$     $$2 \\sqrt{5}$$"}
{"id":"5031","difficulty":"0.6","question_list":[{"type":"text","text":"$$\\triangle D E F$$ is an equilateral triangle. Extend $$F D$$, $$D E$$, $$E F$$ to points $$A$$, $$B$$, $$C$$ respectively, such that $$D A = E B = F C$$. Connect $$A B$$, $$A C$$, $$B C$$. Connect $$B F$$ and extend it to intersect $$A C$$ at point $$G$$. If $$A D = D F = 2$$, then $$\\angle D B F = \\underset{\\underline}{}$$   , $$F G = \\underset{\\underline}{}$$      ."},{"type":"image_path","image_path":"images/5031_q0.png"}],"answer":"$$30 \\circ$$     $$\\frac{4}{5} \\sqrt{3}$$/$$\\frac{4 \\sqrt{3}}{5}$$"}
{"id":"5032","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\bigodot O$$, points A and B lie on $$\\bigodot O$$, $$\\angle A O B = 90 \\circ$$, $$O A = 6$$, point C lies on OA such that $$O C = 2 A C$$, point D is the midpoint of OB, and point M is a moving point on the minor arc AB. Then the minimum value of $$C M + 2 D M$$ is   ."},{"type":"image_path","image_path":"images/5032_q0.png"}],"answer":"$$4 \\sqrt{10}$$"}
{"id":"5038","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, points E and F are points on the sides of square $$A B C D$$, $$C E \\bot D F$$, points G and H are the midpoints of segments $$C E$$ and $$D F$$ respectively, connect $$G H$$. If $$C F = 2$$, $$G H = 3 \\sqrt{2}$$, then the length of $$A B$$ is             ."},{"type":"image_path","image_path":"images/5038_q0.png"}],"answer":"8"}
{"id":"5065","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2, A D = \\sqrt{7}$$, point $$P$$ moves along the sides of the rectangle in the order $$B \\rightarrow C \\rightarrow D \\rightarrow A$$. When point $$P$$ does not coincide with points $$A$$ or $$B$$, fold $$\\triangle A B P$$ along $$A P$$ to obtain $$\\triangle A B^{'} P$$, and connect $$C B^{'}$$. Then, during the motion of point $$P$$, the minimum value of segment $$C B^{'}$$ is          ."},{"type":"image_path","image_path":"images/5065_q0.png"}],"answer":"$$\\sqrt{11} - 2$$/$$- 2 + \\sqrt{11}$$"}
{"id":"5073","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C = 35 \\circ$$. Rotate $$\\triangle A B C$$ around point A by $$\\alpha \\left(\\right. 0 \\circ < \\alpha < 180 \\circ \\left.\\right)$$, such that the rotated point B falls on $$B C$$, and the corresponding point of B is D. Connect $$A D$$, and $$A D$$ is the angle bisector of $$\\angle B A C$$. Then $$\\alpha =$$        ."},{"type":"image_path","image_path":"images/5073_q0.png"}],"answer":"$$\\left(\\frac{110}{3}\\right) \\circ$$"}
{"id":"5104","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, point E is a point outside square ABCD, ∠AEB = 90°. Rotate right triangle ABE counterclockwise around point A by 90° to obtain triangle ADF. Extend DF to intersect BE at point H. If BH = 7 and BC = 13, then DH =     ."},{"type":"image_path","image_path":"images/5104_q0.png"}],"answer":"17"}
{"id":"5109","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$AB = 4$$, $$AC = 6$$, $$E$$ is the midpoint of $$BC$$, $$AD$$ is the angle bisector of $$\\triangle ABC$$, the area of $$\\triangle ABC$$ is denoted as $$S_{1}$$, and the area of $$\\triangle ADE$$ is denoted as $$S_{2}$$, then $$S_{2} : S_{1} =$$      ."},{"type":"image_path","image_path":"images/5109_q0.png"}],"answer":"$$1 : 10$$"}
{"id":"5110","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, construct a regular pentagon $ABCDE$ and an equilateral triangle $ABF$ on the same side of $AB$. Connect $FE$ and $FC$. Then the measure of $\\angle EFA$ is            ."},{"type":"image_path","image_path":"images/5110_q0.png"}],"answer":"66°"}
{"id":"5120","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = \\sqrt{2}$$, rotate $$\\triangle A B C$$ counterclockwise around point B by $$45 \\circ$$ to obtain $$\\triangle A_{1} B C_{1}$$, then the area of the shaded region is      ."},{"type":"image_path","image_path":"images/5120_q0.png"}],"answer":"$$\\frac{\\sqrt{2}}{2}$$"}
{"id":"5128","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the rhombus $$A B C D$$ has a side length of $$6 \\text{cm}$$, $$\\angle B A D = 60 \\circ$$, and when the rhombus is translated along the direction of AC by $$2 \\sqrt{3} \\text{cm}$$, it forms quadrilateral $$A ' B ' C ' D '$$, and $$A ' D '$$ intersects CD at point E. The distance from point E to AC is            $$\\text{cm}$$."},{"type":"image_path","image_path":"images/5128_q0.png"}],"answer":"2"}
{"id":"5145","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$ with side length 4, point $$E$$ is a point on diagonal $$BD$$. If the circumcircle $$\\bigodot O$$ of $$\\triangle ABE$$ is tangent to side $$CD$$, then the radius of $$\\bigodot O$$ is          ."},{"type":"image_path","image_path":"images/5145_q0.png"}],"answer":"$$\\frac{5}{2}$$"}
{"id":"5161","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, points G and E lie on sides $$B C$$ and $$D C$$ respectively. Connect $$A G$$, $$E G$$, and $$A E$$. Fold $$\\triangle A B G$$ and $$\\triangle E C G$$ along $$A G$$ and $$E G$$ respectively, such that points B and C coincide at the same point on $$A E$$, denoted as point F. If $$C E = 3$$ and $$C G = 4$$, then $$\\sin \\angle D A E =$$       ."},{"type":"image_path","image_path":"images/5161_q0.png"}],"answer":"$$\\frac{7}{25}$$"}
{"id":"5167","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, $$O$$ is the circumcenter of $$\\triangle A B C$$, and the circumradius is $$2 \\sqrt{3}$$. Arcs are drawn with centers at $$A$$, $$B$$, and $$C$$, and radii $$A O$$, $$B O$$, and $$C O$$, intersecting the three sides of $$\\triangle A B C$$ at points $$H$$, $$I$$, $$D$$, $$E$$, $$F$$, and $$G$$, respectively. The area of the shaded region is      ."},{"type":"image_path","image_path":"images/5167_q0.png"}],"answer":"$$6 \\pi - 9 \\sqrt{3}$$"}
{"id":"5185","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in a grid diagram composed of 4 identical small squares, ∠1 + ∠2 + ∠3 =        degrees."},{"type":"image_path","image_path":"images/5185_q0.png"}],"answer":"135"}
{"id":"5189","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle A = 36 \\circ$$, if $$B D$$ is the angle bisector of $$\\triangle A B C$$, then the measure of $$\\angle B D C$$ is     ."},{"type":"image_path","image_path":"images/5189_q0.png"}],"answer":"$$72 \\circ$$"}
{"id":"5196","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, an arc is drawn with center $$A$$ and radius $$A B$$, intersecting $$B C$$ at point $$E$$. Connect $$A E$$, $$D E$$, and $$A C$$. If $$A C \\bot A B$$, $$A B = 6$$, $$A D = 10$$, then the length of $$D E$$ is      ."},{"type":"image_path","image_path":"images/5196_q0.png"}],"answer":"8"}
{"id":"5201","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, the perpendicular bisector MN of BC intersects AB at point D, and CD bisects ∠ACB. If AD = 2 and BD = 3, then the length of AC is     ."},{"type":"image_path","image_path":"images/5201_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"5209","difficulty":"0.6","question_list":[{"type":"text","text":"Given rectangle $$A B C D$$, point $$E$$ is on side $$A D$$, with $$D E < A E$$, connect $$B E$$, point $$G$$ is on side $$B C$$, connect $$E G$$, $$B E$$ bisects $$\\angle A E G$$, if $$B G = 5 G C$$, $$D E = 2 C G$$, $$B E = 2 \\sqrt{10}$$, then the area of $$\\triangle A B E$$ is           ."},{"type":"image_path","image_path":"images/5209_q0.png"}],"answer":"$$4 \\sqrt{6}$$"}
{"id":"5214","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the regular hexagon $$A B C D E F$$ with side length 2, point $$P$$ lies on $$A F$$. A beam of light emanates from point $$P$$, strikes point $$Q$$ on the mirror surface $$B C$$, reflects, and then hits point $$G$$ on $$D E$$. Given that $$P Q \\parallel A B$$ and $$Q G \\parallel C D$$, find $$P Q + Q G =$$           ."},{"type":"image_path","image_path":"images/5214_q0.png"}],"answer":"6"}
{"id":"5226","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in heptagon $$A B C D E F G$$, the extensions of $$A B$$ and $$E D$$ intersect at point O, and the sum of the exterior angles $$\\angle 1$$, $$\\angle 2$$, $$\\angle 3$$, $$\\angle 4$$ is $$220^\\circ$$. Then the measure of $$\\angle B O D$$ is       $$^\\circ$$."},{"type":"image_path","image_path":"images/5226_q0.png"}],"answer":"40"}
{"id":"5235","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, points $$E$$, $$F$$, and $$G$$ lie on sides $$AD$$, $$BC$$, and $$CD$$ respectively, and $$BG \\bot EF$$, with foot of perpendicular at $$H$$. Among the following conclusions: ① $$H$$ is the midpoint of segment $$BG$$; ② $$\\angle DEF = \\angle BGC$$; ③ $$BG = EF$$; ④ $$AE + FC = DG$$, the correct conclusions are        ."},{"type":"image_path","image_path":"images/5235_q0.png"}],"answer":"②③④"}
{"id":"5239","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$. Point $$E$$ lies on segment $$A B$$. Connect $$D E$$ and $$C E$$. Fold $$\\triangle D A E$$ along $$D E$$ such that point $$A$$ coincides exactly with point $$O$$. If $$A D = 2 \\sqrt{3}$$, then the length of $$C E$$ is      ."},{"type":"image_path","image_path":"images/5239_q0.png"}],"answer":"$$2 \\sqrt{7}$$"}
{"id":"5242","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 8, A D = 6$$, points E and F are moving points on sides $$A D$$ and $$C D$$ respectively. Connect $$B E$$ and $$E F$$. Point G is the midpoint of $$B E$$, and point H is the midpoint of $$E F$$. Connect $$G H$$. What is the maximum value of $$G H$$?"},{"type":"image_path","image_path":"images/5242_q0.png"}],"answer":"5"}
{"id":"5287","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$OA$$, $$OB$$ are radii of $$\\bigodot O$$, point C lies on $$\\bigodot O$$, $$\\angle AOB = 30^\\circ$$, $$\\angle OBC = 40^\\circ$$, then $$\\angle OAC =$$      $$^\\circ$$."},{"type":"image_path","image_path":"images/5287_q0.png"}],"answer":"25"}
{"id":"5294","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A D = 12$$, and the circle $$\\bigodot O$$ with diameter $$A D$$ is tangent to $$B C$$ at point $$E$$. Connect $$O C$$. If $$O C = A B$$, then the perimeter of $$\\square A B C D$$ is            ."},{"type":"image_path","image_path":"images/5294_q0.png"}],"answer":"$$24 + 6 \\sqrt{5}$$"}
{"id":"5296","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E and F are the midpoints of sides $$AB$$ and $$BC$$ respectively, connect $$AF$$ and $$DE$$, G and H are the midpoints of $$DE$$ and $$AF$$ respectively, connect $$GH$$, then the length of $$GH$$ is"},{"type":"image_path","image_path":"images/5296_q0.png"}],"answer":"$$\\sqrt{2}$$"}
{"id":"5300","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, a 30° triangle ruler and a straightedge are stacked together. If $$\\angle 1 = 72^\\circ$$, then the measure of $$\\angle 2$$ is      ."},{"type":"image_path","image_path":"images/5300_q0.png"}],"answer":"$$132 \\circ$$"}
{"id":"5340","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a regular quadrilateral inscribed in $$\\bigodot O$$. With points A and O as centers, arcs are drawn with a fixed radius greater than $$\\frac{1}{2} O A$$; these arcs intersect at points M and N. Draw line MN, which intersects $$\\bigodot O$$ at points E and F. If $$O A = 1$$, then the area of the shaded region bounded by $$\\overset{⌢}{B E}$$, $$A E$$, and $$A B$$ is       ."},{"type":"image_path","image_path":"images/5340_q0.png"}],"answer":"$$\\frac{\\pi}{12} + \\frac{\\sqrt{3}}{4} - \\frac{1}{2}$$"}
{"id":"5356","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$ABCD$$, an arc with center $$C$$ and radius $$CD$$ intersects $$AB$$ at point $$E$$, and an arc with center $$C$$ and radius $$CB$$ intersects $$CD$$ at point $$F$$. Connect $$AC$$. If $$AC = \\sqrt{5}$$ and $$BC = 1$$, then the area of the shaded region in the figure is      (express the result in terms of $$\\pi$$)"},{"type":"image_path","image_path":"images/5356_q0.png"}],"answer":"$$\\frac{\\sqrt{3}}{2} + \\frac{1}{12} \\pi$$"}
{"id":"5376","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = 7$$, $$B C = 8$$, $$s i n B = \\frac{4}{5}$$. Point $$P$$ lies on side $$A B$$, $$A P = 2$$, and a circle $$\\bigodot P$$ is drawn with center $$P$$ and radius $$A P$$. Point $$Q$$ lies on side $$B C$$, and a circle $$\\bigodot Q$$ is drawn with center $$Q$$ and radius $$C Q$$. If $$\\bigodot P$$ and $$\\bigodot Q$$ are externally tangent, then the length of $$C Q$$ is        ."},{"type":"image_path","image_path":"images/5376_q0.png"}],"answer":"$$\\frac{37}{14}$$/$$2 \\frac{9}{14}$$"}
{"id":"5382","difficulty":"0.6","question_list":[{"type":"text","text":"In Rt△ABC, ∠C=90°, AD bisects ∠CAB, AC=6, BC=8, CD=       ."},{"type":"image_path","image_path":"images/5382_q0.png"}],"answer":"3."}
{"id":"5388","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the regular pentagon $$A B C D E$$, $$A D$$ and $$C E$$ intersect at point F, and connect $$B F$$. Then the measure of $$\\angle C F B$$ is       ."},{"type":"image_path","image_path":"images/5388_q0.png"}],"answer":"$$54 \\circ$$"}
{"id":"5413","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 3$$, $$B C = 4$$, point D lies on side $$A B$$, $$A D = A C$$, $$A E \\bot C D$$, with foot at F, intersecting $$B C$$ at point E. Find the length of $$B E$$:"},{"type":"image_path","image_path":"images/5413_q0.png"}],"answer":"$$\\frac{5}{2}$$"}
{"id":"5424","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the diameter of $$\\bigodot O$$, and $$\\triangle ABC$$ is an inscribed triangle of $$\\bigodot O$$. If $$\\angle DAC = \\angle ABC$$, and $$AC = 5$$, then $$AD =$$      ."},{"type":"image_path","image_path":"images/5424_q0.png"}],"answer":"$$5 \\sqrt{2}$$"}
{"id":"5431","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the cross-section of the high-speed railway tunnel is part of a circle with center O. The road surface AB = 24 meters, and the clear height CD = 18 meters. Then the length of OD is ______."},{"type":"image_path","image_path":"images/5431_q0.png"}],"answer":"5"}
{"id":"5465","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral ABCD is a rhombus, $$\\angle ABC = 70^\\circ$$, extend BC to E, and within $$\\angle DCE$$, draw ray CM such that $$\\angle ECM = 15^\\circ$$. From point D, draw $$DF \\bot CM$$, with foot at F. If $$DF = \\sqrt{6}$$, then the length of diagonal BD is      ."},{"type":"image_path","image_path":"images/5465_q0.png"}],"answer":"$$2 \\sqrt{6}$$"}
{"id":"5485","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, point E is the midpoint of $$OB$$, and a chord $$CD \\bot AB$$ is drawn through point E. Connect $$AC$$ and $$AD$$. If point F is the midpoint of $$\\overset{⌢}{AC}$$, and a perpendicular $$CG \\bot AF$$ is drawn from point C with foot at point G. If the radius of $$\\bigodot O$$ is 2, then the length of $$CG$$ is        ."},{"type":"image_path","image_path":"images/5485_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"5507","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point E lies on side $$B C$$, and point F lies on side $$A B$$. Fold $$C D$$ along $$D E$$ so that point C lands at point $$C^{'}$$, with $$D E$$ as the fold line; then fold $$B E$$ along $$E F$$ so that point B lands exactly at point $$B^{'}$$ on segment $$E C^{'}$$, with $$E F$$ as the fold line. If $$C D = 8$$, $$B F = 3$$, and $$B^{'} C^{'} = 2$$, then the length of $$B C$$ is      ."},{"type":"image_path","image_path":"images/5507_q0.png"}],"answer":"10"}
{"id":"5514","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given that line $$a \\parallel b$$, $$\\angle 1 = 85^\\circ$$, $$\\angle 2 = 60^\\circ$$, then $$\\angle 3 =$$ ."},{"type":"image_path","image_path":"images/5514_q0.png"}],"answer":"$$35 \\circ$$"}
{"id":"5516","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$B C = 4$$, $$A C$$ is sufficiently long, points D and E lie on sides $$B C$$ and $$A C$$ respectively, F is the midpoint of $$D E$$, and if $$B D = C E$$, then the minimum length of $$C F$$ is         ."},{"type":"image_path","image_path":"images/5516_q0.png"}],"answer":"$$\\sqrt[]{2}$$"}
{"id":"5565","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is a chord of $$\\bigodot O$$, $$BC$$ is a line passing through point $$B$$, $$\\angle AOB = 130^\\circ$$, when $$\\angle ABC =$$      , $$BC$$ is a tangent to $$\\bigodot O$$."},{"type":"image_path","image_path":"images/5565_q0.png"}],"answer":"$$65 \\circ$$"}
{"id":"5593","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, $$B C = 2 \\sqrt{2}$$, an arc is drawn with center at point A and radius $$A D$$, intersecting $$B C$$ at point E. Connect $$A E$$. The area of the shaded region in the figure is      ."},{"type":"image_path","image_path":"images/5593_q0.png"}],"answer":"$$4 \\sqrt{2} - 2 - \\pi$$"}
{"id":"5629","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given rectangle $$A B C D$$, with $$A B = 4$$, $$A D = 3$$, point E is a moving point on side $$D C$$, not coinciding with either endpoint. Connect $$B E$$, and fold $$B C E$$ along $$B E$$ to obtain $$B E F$$. Connect $$A F$$ and extend it to intersect $$C D$$ at point G. Then the maximum value of segment $$C G$$ is       ."},{"type":"image_path","image_path":"images/5629_q0.png"}],"answer":"$$4 - \\sqrt{7}$$/$$- \\sqrt{7} + 4$$"}
{"id":"5631","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$ABCD$$ with side length 8, $$E$$ is the midpoint of side $$AD$$, and connecting $$CE$$ intersects diagonal $$BD$$ at point $$F$$. If $$\\angle DEF = \\angle DFE$$, then the area of this rhombus is        ."},{"type":"image_path","image_path":"images/5631_q0.png"}],"answer":"$$24 \\sqrt{7}$$"}
{"id":"5650","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, a cylindrical glass cup is placed horizontally, with a cross-section that is a circle of radius $$5 \\text{cm}$$. The width of the water surface inside the cup is $$AB = 8 \\text{cm}$$. The depth of the water $$CD$$ is           ."},{"type":"image_path","image_path":"images/5650_q0.png"}],"answer":"$$2 ( \\text{cm} )$$"}
{"id":"5662","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = A C$$, point $$D$$ is a point on $$A C$$, connect $$B D$$, draw $$A E \\bot B D$$ through point $$A$$, intersecting at point $$E$$, draw $$C F \\bot B D$$ through point $$C$$, intersecting the extension of $$B D$$ at point $$F$$, if $$A E = 3$$, $$C F = 2$$, then the length of $$B F$$ is      ."},{"type":"image_path","image_path":"images/5662_q0.png"}],"answer":"$$8$$"}
{"id":"5672","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A D$$ is the altitude, $$A B > A C$$, E and F are the midpoints of $$A B$$ and $$B C$$ respectively. If $$\\angle C = \\alpha$$, then the measure of $$\\angle D E F$$ is      (expressed in terms of $$\\alpha$$)."},{"type":"image_path","image_path":"images/5672_q0.png"}],"answer":"$$2 \\alpha - 90 \\circ$$"}
{"id":"5687","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 3, and points E, F, G lie on sides $$A B, B C, C D$$ respectively, with $$A F \\bot E G$$. When $$C F = 2 B F$$, the minimum value of $$E F + A G$$ is     ."},{"type":"image_path","image_path":"images/5687_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"5715","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 8$$, $$t a n \\angle B A C = \\frac{3}{2}$$, point $$D$$ is the midpoint of side $$B C$$, point $$E$$ is any point on side $$A B$$ (point $$E$$ does not coincide with point $$B$$). Fold $$\\triangle D B E$$ along $$D E$$ so that point $$B$$ lands at point $$F$$, and connect $$A F$$. When the length of segment $$A F$$ is minimized, the distance between points $$B$$ and $$F$$ is      ."},{"type":"image_path","image_path":"images/5715_q0.png"}],"answer":"$$\\frac{24 \\sqrt{5}}{5}$$"}
{"id":"5723","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, M is the midpoint of side BC in the obtuse triangle ABC. The line MN passing through M divides triangle ABC into two parts with equal perimeters. Given AB = 8 and angle A = 120°, find MN."},{"type":"image_path","image_path":"images/5723_q0.png"}],"answer":"4"}
{"id":"5729","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$ABCD$$, $$AB = 4 \\text{ cm}$$, $$BC = 3 \\text{ cm}$$, point E is the midpoint of $$CD$$, and point P moves from point A at a speed of 1 cm per second along $$A \\rightarrow B \\rightarrow C \\rightarrow E$$, finally reaching point E. If the time taken by point P is $$x$$ seconds, then when $$x =$$        seconds, the area of $$\\triangle APE$$ equals $$5 \\left(\\text{cm}\\right)^2$$."},{"type":"image_path","image_path":"images/5729_q0.png"}],"answer":"$$\\frac{10}{3}$$ or $$5$$"}
{"id":"5733","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, after folding rectangle paper $$ABCD$$ along $$EF$$, point D and point C land at points $$D'$$ and $$C'$$ respectively, and the extension of $$ED'$$ passes exactly through point B. If $$DE = DC = 3$$ and $$CF = 2$$, then $$AE$$ equals           ."},{"type":"image_path","image_path":"images/5733_q0.png"}],"answer":"4"}
{"id":"5740","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$\\angle A O B = 60 \\circ$$, with a circle centered at point $$O$$ intersecting the two sides of the angle at points C and D. With $$O$$ as the center and a radius greater than $$\\frac{1}{2} C D$$, two arcs intersect at a point $$P$$ inside $$\\angle A O B$$. Connect $$O P$$. Draw line $$P E \\parallel O A$$ intersecting $$O B$$ at point $$E$$, and draw line $$P F \\parallel O B$$ intersecting $$O A$$ at point $$F$$. Given $$O P = 6 \\text{cm}$$, the area of quadrilateral $$P F O E$$ is      ."},{"type":"image_path","image_path":"images/5740_q0.png"}],"answer":"$$6 \\sqrt{3}$$"}
{"id":"5741","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a cyclic quadrilateral inscribed in $$\\bigodot O$$, $$D B$$ bisects $$\\angle A D C$$, connect $$O C$$, $$B D$$, $$O C \\bot B D$$, if $$\\angle A$$ equals $$50 \\circ$$, then the measure of $$\\angle A D B$$ is         ."},{"type":"image_path","image_path":"images/5741_q0.png"}],"answer":"$$25 \\circ$$"}
{"id":"5755","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 120 \\circ$$, $$A B = 6$$, $$A C = 4$$, point $$M$$ is a moving point on side $$A B$$, connect $$C M$$, and let $$\\bigodot O$$ with diameter $$A M$$ intersect $$C M$$ at point $$N$$. Then the minimum value of segment $$B N$$ is      ."},{"type":"image_path","image_path":"images/5755_q0.png"}],"answer":"$$2 \\sqrt{13} - 2$$"}
{"id":"5789","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, it is given that $$A B = 4$$, $$B C = 6$$, $$\\angle A B C = 60^\\circ$$. Point P is a moving point on side $$B C$$ (point P does not coincide with B or C). Connect $$A P$$, and construct point Q as the reflection of point B over line $$A P$$. Then the minimum value of segment $$Q C$$ is            ."},{"type":"image_path","image_path":"images/5789_q0.png"}],"answer":"$$2 \\sqrt{7} - 4$$/$$- 4 + 2 \\sqrt{7}$$"}
{"id":"5794","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the radius of $$\\bigodot O$$ is 3, $$AB$$ is the diameter. Arcs are drawn with centers at points $$A$$ and $$B$$ and radius equal to the length of $$AB$$. These arcs intersect at points $$C$$ and $$D$$. The area of the shaded region in the figure is      ."},{"type":"image_path","image_path":"images/5794_q0.png"}],"answer":"$$15 \\pi - 18 \\sqrt{3}$$"}
{"id":"5795","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the parallelogram paper $$A B C D$$ is folded along the diagonal $$A C$$, and point B lands at point $$B^{'}$$, with $$B^{'} C$$ intersecting $$A D$$ at point E. At this time, $$\\triangle C D E$$ is exactly an equilateral triangle. If $$A B = 6 \\text{cm}$$, then $$A D =$$       $$\\text{cm}$$."},{"type":"image_path","image_path":"images/5795_q0.png"}],"answer":"12"}
{"id":"5817","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given isosceles right triangle $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A B = \\sqrt{2}$$, point C is a common vertex of rectangle $$B C G F$$ and $$\\triangle A B C$$, and $$C E = 1$$, $$C G = 3$$; point D is a point on the extension of $$C B$$ such that $$C D = 2$$. Connect $$B G$$ and $$D F$$. During the process in which rectangle $$E C G F$$ rotates clockwise around point C for one full revolution, when segment $$B G$$ reaches its maximum and minimum lengths, the corresponding lengths of segment $$D F$$ are m and n, respectively. Find the value of $$\\frac{m}{n}$$."},{"type":"image_path","image_path":"images/5817_q0.png"}],"answer":"$$\\sqrt{13}$$"}
{"id":"5830","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral ABCD is a rectangle, $$AB = 3$$, $$BC = 4$$, point P is a moving point on segment $$BC$$, point M is a point on segment $$AP$$, $$\\angle ADM = \\angle BAP$$, then the minimum value of $$BM$$ is      ."},{"type":"image_path","image_path":"images/5830_q0.png"}],"answer":"$$\\sqrt{13} - 2$$"}
{"id":"5844","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, point $$C$$ is a point on the circle, $$\\angle BAC = 20^\\circ$$, and the minor arc $$\\overset{⌢}{AC}$$ is reflected across the line containing chord $$AC$$, intersecting $$AB$$ at point $$D$$. Then the measure of $$\\angle ACD$$ is      ."},{"type":"image_path","image_path":"images/5844_q0.png"}],"answer":"$$50 \\circ$$"}
{"id":"5859","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, points E and F lie on sides $$B C$$ and $$C D$$ respectively, $$A E = A F$$, $$\\angle E A F = 30^{\\circ}$$, then $$\\angle A E B =$$       $$\\circ$$."},{"type":"image_path","image_path":"images/5859_q0.png"}],"answer":"$$60$$"}
{"id":"5875","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point O. A line passing through point O intersects $$AD$$ and $$BC$$ at points M and N, respectively. If the area of $$\\triangle CON$$ is 2 and the area of $$\\triangle DOM$$ is 4, then the area of $$\\square ABCD$$ is          ."},{"type":"image_path","image_path":"images/5875_q0.png"}],"answer":"24"}
{"id":"5884","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, extend the sides of the regular pentagon $$F G H J K$$ such that $$G A = H B = J C = K D = F E$$; if $$A B = A F$$, then the measure of $$\\angle B C H$$ is         ."},{"type":"image_path","image_path":"images/5884_q0.png"}],"answer":"$$36 \\circ$$"}
{"id":"5891","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C$$, $$\\angle A C B = 90 \\circ$$. D, E, F are points on sides $$A B$$, $$A C$$, $$B C$$ respectively, and $$C E = C F$$. If $$A D = 2$$, $$B D = 1$$, then the minimum value of $$D E + D F$$ is          ."},{"type":"image_path","image_path":"images/5891_q0.png"}],"answer":"$$\\sqrt{5}$$"}
{"id":"5893","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 2$$, D is the midpoint of $$A B$$, and $$\\triangle B C D$$ is folded along the line $$C D$$, with point B landing at point E. If $$C E \\bot A B$$, then $$B E =$$         ."},{"type":"image_path","image_path":"images/5893_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"5918","difficulty":"0.6","question_list":[{"type":"text","text":"Point F is the midpoint of side DE of the regular pentagon $$A B C D E$$, connect $$B F$$ and extend it to intersect the extension of $$C D$$ at point G, then the measure of $$\\angle B G C$$ is      ."},{"type":"image_path","image_path":"images/5918_q0.png"}],"answer":"$$18 \\circ$$"}
{"id":"5932","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, point $$E$$ lies on side $$B C$$, $$B E = \\frac{3}{2}$$, and an isosceles right triangle $$A E F$$ is constructed with $$\\angle A E F = 90^\\circ$$."},{"type":"image_path","image_path":"images/5932_q0.png"},{"type":"text","text":"(1) The length of $$C F$$ is      . (2) If $$M$$ is the midpoint of $$A F$$, and segment $$D M$$ is connected, then the length of $$D M$$ is      ."}],"answer":"$$\\frac{3 \\sqrt{2}}{2}$$     $$\\frac{5 \\sqrt{2}}{4}$$"}
{"id":"5934","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the rhombus $$A B C D$$ has side length 6, $$\\angle B A D = 120 \\circ$$, and from point $$D$$, draw $$D E \\bot B C$$, intersecting the extension of $$B C$$ at point $$E$$. Connect $$A E$$, intersecting $$B D$$ and $$C D$$ at points $$F$$ and $$G$$, respectively. Then the length of $$F G$$ is      ."},{"type":"image_path","image_path":"images/5934_q0.png"}],"answer":"$$\\frac{4 \\sqrt{7}}{5}$$/$$\\frac{4}{5} \\sqrt{7}$$"}
{"id":"5953","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$PA$$ is tangent to $$\\bigodot O$$ at point A, $$PO$$ intersects $$\\bigodot O$$ at point B, point C lies on $$PA$$, and $$CB = CA$$. If $$OA = 5$$, $$PA = 12$$, then the length of $$CA$$ is      ."},{"type":"image_path","image_path":"images/5953_q0.png"}],"answer":"$$\\frac{10}{3}$$"}
{"id":"5959","difficulty":"0.6","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, $$\\angle A B C = 60 \\circ$$, $$A B = 9$$, point D is a point on side AB, $$B D = B C$$, connect CD, fold $$\\triangle A D C$$ along CD to obtain $$\\triangle A_{1} D C$$, where $$A_{1} C$$ intersects side AB at point E, $$B E = 4$$, connect $$A_{1} B$$, then the length of $$A_{1} B$$ is      ."},{"type":"image_path","image_path":"images/5959_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"5963","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C E$$ is inscribed in $$\\bigodot O$$, connect $$A C$$, $$A C$$ is the diameter of $$\\bigodot O$$, $$E$$ is the midpoint of $$\\overset{⌢}{A C B}$$. Draw the tangent $$E F$$ to $$\\bigodot O$$ through point $$E$$, intersecting the extension of $$B C$$ at point $$F$$, and $$E F \\bot B C$$, $$E F = 4$$, $$B F = 5$$, then the length of $$A E$$ is    , and the radius of $$\\bigodot O$$ is       ."},{"type":"image_path","image_path":"images/5963_q0.png"}],"answer":"$$\\sqrt{41}$$     $$\\frac{41}{10}$$"}
{"id":"5975","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, hexagon $$A B C D E F$$ is a regular hexagon inscribed in $$\\bigodot O$$. Let the area of the regular hexagon $$A B C D E F$$ be $$S_{1}$$, and the area of $$\\triangle A C E$$ be $$S_{2}$$. Then $$\\frac{S_{1}}{S_{2}} =$$         ."},{"type":"image_path","image_path":"images/5975_q0.png"}],"answer":"2"}
{"id":"5985","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$ and $$\\triangle A D E$$, $$A B = A C$$, $$\\angle B A C = \\angle D A E = 40 \\circ$$. Rotate $$\\triangle A D E$$ clockwise around point A by a certain angle. When $$A D \\bot B C$$, the measure of $$\\angle B A E$$ is      ."},{"type":"image_path","image_path":"images/5985_q0.png"}],"answer":"$$60 \\circ$$ or $$120 \\circ$$"}
{"id":"5987","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the square $$ABCD$$ with side length 6, point M is the midpoint of $$AB$$, and point E lies on $$AD$$ such that $$AE = \\frac{1}{3} AD$$. In the isosceles triangle $$EDF$$, $$ED = FD$$, and $$\\angle EDF = 120^\\circ$$."},{"type":"image_path","image_path":"images/5987_q0.png"},{"type":"text","text":"(1) The area of $$\\triangle EDF$$ is      ; (2) If N is the midpoint of $$EF$$, then the value of $$MN^2$$ is      ."}],"answer":"$$4 \\sqrt{3}$$     $$37 - 6 \\sqrt{3}$$/$$- 6 \\sqrt{3} + 37$$"}
{"id":"5988","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in triangle paper $$A B C$$, $$A B = A C, \\angle B = 20 \\circ$$, point $$D$$ is a moving point on side $$B C$$, fold the triangle paper along $$A D$$ so that point $$B$$ lands at point $$B^{'}$$, when $$B^{'} D \\bot B C$$, the degree measure of $$\\angle B A D$$ is           ."},{"type":"image_path","image_path":"images/5988_q0.png"}],"answer":"$$25 \\circ$$ or $$115 \\circ$$"}
{"id":"5993","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C = 6$$, $$B D = 8$$, and from point A, draw $$A E \\bot C D$$ intersecting at point E, then $$A E$$ is      ."},{"type":"image_path","image_path":"images/5993_q0.png"}],"answer":"$$\\frac{2 \\text{4}}{\\text{5}}$$"}
{"id":"6025","difficulty":"0.6","question_list":[{"type":"text","text":"An exterior angle of a triangle is 100°, then the angle formed by the bisectors of the two non-adjacent interior angles (obtuse angle) is      ."},{"type":"image_path","image_path":"images/6025_q0.png"}],"answer":"130°"}
{"id":"6033","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, and point $$C$$ is a point on $$\\bigodot O$$. The minor arc $$BC$$ is folded along chord $$BC$$ and intersects $$AB$$ at point $$D$$. Point $$E$$ is the midpoint of arc $$ACB$$. If $$AC = \\sqrt{10}$$ and $$AD = 2$$, then the radius of $$\\bigodot O$$ is $$r =$$        ; the area of $$\\triangle BCE$$ is        ."},{"type":"image_path","image_path":"images/6033_q0.png"}],"answer":"5     15"}
{"id":"6048","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 6, AD = 4. Points E and F are moving points on AB and DC respectively, with EF ∥ BC. The minimum value of AF + CE is      ."},{"type":"image_path","image_path":"images/6048_q0.png"}],"answer":"10"}
{"id":"6051","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in triangle $$A B C$$, $$\\angle A C B = 90 \\circ$$, $$A B = 10$$, $$A C = 8$$, points $$D$$ and $$E$$ are points on segments $$A C$$ and $$A B$$ respectively, and connect $$D E$$. Fold $$\\triangle A D E$$ along $$D E$$ so that point $$A$$ lands at point $$F$$ on the extension of $$B C$$, at which time $$\\angle B F E = 30 \\circ$$, then the length of $$C F$$ is      ."},{"type":"image_path","image_path":"images/6051_q0.png"}],"answer":"$$\\frac{40 \\sqrt{3} - 48}{13}$$"}
{"id":"6067","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, point E lies on side BC. Triangle CDE is folded along DE to obtain triangle FDE, with point F landing on AE. If CE = 3 cm and AF = 2 EF, then AB =      cm."},{"type":"image_path","image_path":"images/6067_q0.png"}],"answer":"$$3 \\sqrt{5}$$"}
{"id":"6079","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AC$$ bisects $$\\angle DCB$$, $$CB = CD$$, the extension of $$DA$$ intersects $$BC$$ at point $$E$$, if $$\\angle EAC = 49^{\\circ}$$, then the measure of $$\\angle BAE$$ is          ."},{"type":"image_path","image_path":"images/6079_q0.png"}],"answer":"$$82 \\circ .$$"}
{"id":"6107","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, on the side $$AD$$ of the square $$ABCD$$, there is a point $$E$$. Connect $$BE$$, and from point $$E$$, draw $$EF \\bot BE$$ (point $$F$$ lies to the right of side $$CD$$), with foot at point $$E$$. $$EF$$ intersects $$CD$$ at point $$G$$. Connect $$DF$$. If $$\\angle CDF = 45^\\circ$$, point $$G$$ is the midpoint of $$EF$$, and $$DG = 1$$."},{"type":"image_path","image_path":"images/6107_q0.png"},{"type":"text","text":"(Ⅰ) The length of segment $$DE$$ is     ; (Ⅱ) The length of segment $$BE$$ is     ."}],"answer":"2     $$2 \\sqrt{5}$$"}
{"id":"6110","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E is the midpoint of $$AB$$, connect $$DE$$, rotate $$\\triangle DAE$$ counterclockwise about point D by $$90^\\circ$$ to obtain $$\\triangle DCF$$, connect $$EF$$, then the length of $$EF$$ is      ."},{"type":"image_path","image_path":"images/6110_q0.png"}],"answer":"$$2 \\sqrt{10}$$"}
{"id":"6116","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, semicircle $$O$$, with point $$O$$ as the center and diameter $$AB$$ of length 6, draw an arc with point $$B$$ as the center and $$OB$$ as the radius, intersecting arc $$AB$$ at point $$C$$. The area of the shaded region is      ."},{"type":"image_path","image_path":"images/6116_q0.png"}],"answer":"$$\\frac{3}{2} \\pi + \\frac{9}{4} \\sqrt{3}$$"}
{"id":"6117","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in sector $$B O C$$, $$\\angle B O C = 60^\\circ$$, $$O D$$ bisects $$\\angle B O C$$ and intersects arc $$B C$$ at point $$D$$. Point $$E$$ is a moving point on radius $$O B$$. If $$O B = 2$$, then the minimum value of the perimeter of the shaded region is          ."},{"type":"image_path","image_path":"images/6117_q0.png"}],"answer":"$$2 \\sqrt{2} + \\frac{\\pi}{3} .$$"}
{"id":"6120","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A = 60 \\circ$$, $$A C = 2$$, point $$D$$ is the midpoint of side $$A B$$, and an arc $$B C$$ is drawn with point $$D$$ as the center and $$B D$$ as the radius. The area of the shaded region in the figure is      ."},{"type":"image_path","image_path":"images/6120_q0.png"}],"answer":"$$\\frac{4 \\pi}{3} - \\sqrt{3}$$"}
{"id":"6131","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$C A = C B$$, $$\\angle A C B = 90 \\circ$$, $$A B = 2$$, point $$D$$ is the midpoint of $$A B$$, and a sector $$E D F$$ with center $$D$$ and central angle $$90 \\circ$$ is drawn; point $$C$$ lies exactly on $$\\overset{⌢}{E F}$$. The area of the shaded region in the figure is      ."},{"type":"image_path","image_path":"images/6131_q0.png"}],"answer":"$$\\frac{\\pi}{4} - \\frac{1}{2}$$"}
{"id":"6133","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, the circle $$\\bigodot O$$ with diameter AB intersects BC at point D. Draw the tangent DE to $$\\bigodot O$$ through point D, intersecting the extension of AB at point E. If $$A B = 4$$ and $$\\tan \\angle C A B = 2$$, then the length of the tangent DE is      ."},{"type":"image_path","image_path":"images/6133_q0.png"}],"answer":"$$4$$"}
{"id":"6145","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, point $$E$$ lies on the extension of $$CB$$, connect $$AE$$, point $$F$$ is the midpoint of $$AE$$, connect $$OF$$ intersecting $$AB$$ at point $$G$$, connect $$BF$$. If $$BE = 3$$, $$OF = \\frac{9}{2}$$, then the length of $$BF$$ is      ."},{"type":"image_path","image_path":"images/6145_q0.png"}],"answer":"$$\\frac{3 \\sqrt{5}}{2}$$"}
{"id":"6148","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given square $$A B C D$$, point $$E$$ is the midpoint of $$A B$$, connect $$D E$$. $$E F \\bot D E$$ intersects $$B C$$ at point $$K$$, and $$E F = D E$$, connect $$D F$$ intersecting $$B C$$ at point $$H$$. $$F G \\bot A B$$ intersects the extension of $$A B$$ at point $$G$$. Find the value of $$C H : H K : B K$$ is      ."},{"type":"image_path","image_path":"images/6148_q0.png"}],"answer":"$$4 : 5 : 3$$"}
{"id":"6152","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point O, and $$O A = O B$$, $$\\angle O A D = 65^\\circ$$, then $$\\angle O D C =$$        ."},{"type":"image_path","image_path":"images/6152_q0.png"}],"answer":"$$25 \\circ$$"}
{"id":"6180","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, a square $ABCD$ with side length $\\sqrt{2}$ is inscribed in $\\bigodot O$. Tangents to $\\bigodot O$ are drawn through points $A$ and $D$, intersecting at point $P$. The area of the shaded region in the figure is        ."},{"type":"image_path","image_path":"images/6180_q0.png"}],"answer":"$$1 - \\frac{\\pi}{4}$$"}
{"id":"6182","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ, \\angle A = 30 \\circ, A B = 8$$. If point D lies on line $$A B$$ (not coinciding with points A or B), and $$\\angle B C D = 30 \\circ$$, then the length of $$A D$$ is             ."},{"type":"image_path","image_path":"images/6182_q0.png"}],"answer":"6 or 12"}
{"id":"6189","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A B = 2 A C = 4$$, $$C O$$ is the median to the hypotenuse, point P is a moving point on segment $$A O$$, rotate segment $$P C$$ counterclockwise by $$90 \\circ$$ about point P to obtain segment $$P Q$$, connect $$C Q$$ and $$O Q$$, when $$P C$$ is perpendicular to one side of $$\\triangle A B C$$, the value of segment $$O Q$$ is          ."},{"type":"image_path","image_path":"images/6189_q0.png"}],"answer":"$$\\sqrt{3} - 1$$ or $$\\sqrt{6} - \\sqrt{2}$$"}
{"id":"6190","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$\\triangle A B C$$, the base $$B C = 5$$, D is a point on the leg $$A B$$, and $$C D = 4$$, $$B D = 3$$, then the length of $$A D$$ is       ."},{"type":"image_path","image_path":"images/6190_q0.png"}],"answer":"$$\\frac{7}{6}$$/$$1 \\frac{1}{6}$$"}
{"id":"6195","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$\\angle C = 120^\\circ$$, $$A B = 2$$, $$A D = 2 A B$$, points H and G are moving points on sides $$D C$$ and $$B C$$ respectively. Connect $$A H$$ and $$H G$$. Point E is the midpoint of $$A H$$, and point F is the midpoint of $$G H$$. Connect $$E F$$. What is the minimum value of $$E F$$?"},{"type":"image_path","image_path":"images/6195_q0.png"}],"answer":"$$\\frac{\\sqrt{3}}{2}$$/$$0 . 5 \\sqrt{3}$$/$$\\frac{1}{2} \\sqrt{3}$$"}
{"id":"6240","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, BE is the median of △ABC, point D is a point on side BC, BD = 2CD, BE and AD intersect at point F. If the area of △ABC is 24, then S△BDF - S△AEF equals     ."},{"type":"image_path","image_path":"images/6240_q0.png"}],"answer":"4"}
{"id":"6250","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of the inscribed regular hexagon $ABCDEF$ in circle $\\bigodot O$ is 4, and H is the midpoint of side $AF$. The area of the shaded region in the figure is ______."},{"type":"image_path","image_path":"images/6250_q0.png"}],"answer":"$$\\text{4} \\sqrt{3} + \\frac{8}{3} \\pi$$"}
{"id":"6264","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given that the chord $$AB = 6$$ of $$\\bigodot O$$, construct a square $$ABCD$$ with $$AB$$ as one side, and side $$CD$$ is tangent to $$\\bigodot O$$ at point $$E$$. Then the radius of $$\\bigodot O$$ is"},{"type":"image_path","image_path":"images/6264_q0.png"}],"answer":"$$\\frac{15}{4}$$"}
{"id":"6270","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$A B C$$, $$A B = A C = \\sqrt{5}, B C = 2, A D$$ bisects $$\\angle B A C, G E$$ is the perpendicular bisector of $$A C$$ intersecting $$A D$$ at point $$F$$, then the length of $$A F$$ is      ."},{"type":"image_path","image_path":"images/6270_q0.png"}],"answer":"$$\\frac{5}{4}$$"}
{"id":"6284","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, point E lies on $$A D$$, and $$E C$$ bisects $$\\angle B E D$$. If $$\\angle E B C = 30^\\circ$$ and $$B E = 10$$, then the area of $$\\square A B C D$$ is        ."},{"type":"image_path","image_path":"images/6284_q0.png"}],"answer":"50"}
{"id":"6296","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, ∠ADC = 90°, AC ⊥ BC, ∠ABC = 45°, AC intersects BD at point E. If AB = $$2 \\sqrt{10}$$, CD = 2, then the area of △ABE is         ."},{"type":"image_path","image_path":"images/6296_q0.png"}],"answer":"$$\\frac{60}{7}$$"}
{"id":"6297","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given that the side length of square ABCD is 5, points E and F lie on AD and DC respectively, with AE = DF = 2. BE intersects AF at point G. Point H is the midpoint of BF. Connect GH; then the length of GH is       ."},{"type":"image_path","image_path":"images/6297_q0.png"}],"answer":"$$\\frac{\\sqrt{34}}{2}$$"}
{"id":"6319","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, point O lies on side $$A C$$, and a circle with center O and radius 3 passes exactly through point C and is tangent to side $$A B$$ at point D, intersecting side $$B C$$ at point E. The length of the minor arc $$D E$$ is       (answer in terms of π)."},{"type":"image_path","image_path":"images/6319_q0.png"}],"answer":"$$\\frac{3}{2} \\pi$$"}
{"id":"6321","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is inscribed in a circle $$\\bigodot O$$ with radius $$\\frac{7 \\sqrt{3}}{3}$$, $$B D \\bot A C$$ at point $$D$$, the extension of $$B D$$ intersects $$\\bigodot O$$ at point $$E$$, $$M$$ is the midpoint of $$\\overset{⌢}{A B E}$$, connect $$O M$$ intersecting $$A C$$ at point $$F$$, if $$A C = 8$$, $$\\angle C = 60 \\circ$$, $$A B > 2 B C$$, then $$D E =$$      , $$O F =$$         ."},{"type":"image_path","image_path":"images/6321_q0.png"}],"answer":"$$\\frac{13 \\sqrt{3}}{6}$$     $$\\frac{2}{3}$$"}
{"id":"6322","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, △ABC is an equilateral triangle, AD is the altitude from A to side BC, E is the midpoint of AC, and P is a moving point on AD. When the sum of PC and PE is minimized, the measure of ∠CPE is        °."},{"type":"image_path","image_path":"images/6322_q0.png"}],"answer":"60"}
{"id":"6328","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of the equilateral triangle ABC is 4, the radius of $$\\bigodot C$$ is $$\\sqrt{3}$$, and P is a moving point on side AB. Draw the tangent PQ to $$\\bigodot C$$ from point P, with Q as the point of tangency. The minimum value of PQ is        ."},{"type":"image_path","image_path":"images/6328_q0.png"}],"answer":"3"}
{"id":"6352","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, point E lies on $$A B$$, connect $$C E$$ and $$D E$$, draw $$D F \\bot C E$$ through point D, with foot at F. If $$C E = C D$$, $$D F = 3$$, $$B E = 4$$, then $$D E =$$        ."},{"type":"image_path","image_path":"images/6352_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"6360","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A O B$$, $$O B = 2 \\sqrt{3}, \\angle A = 30 \\circ, \\bigodot O$$ has a radius of $$1,$$ point $$P$$ is a moving point on side $$A B$$, and from point $$P$$, draw a tangent $$P Q$$ to $$\\bigodot O$$ (with point $$Q$$ as the point of tangency). Then the minimum length of segment $$P Q$$ is    ."},{"type":"image_path","image_path":"images/6360_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"6405","difficulty":"0.6","question_list":[{"type":"text","text":"In $$\\text{Rt} \\triangle A B C$$, $$\\angle B = 90 \\circ$$, $$A B = 6$$, $$\\sin C = \\frac{3}{5}$$, D is a moving point on side $$A B$$, rotate $$D A$$ about point D so that point A lands at point E on side $$A C$$, draw $$E F \\bot D E$$ intersecting side $$B C$$ at point F, connect $$D F$$, when $$\\triangle D E F$$ is an isosceles triangle, the length of segment $$C F$$ is      ."},{"type":"image_path","image_path":"images/6405_q0.png"}],"answer":"$$\\frac{25}{7}$$"}
{"id":"6412","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, square $$A B C D$$, E and F are the midpoints of $$A B$$ and $$B C$$ respectively, $$A F$$ and $$D E$$ intersect at point G, connect $$C G$$, if $$A B = 2$$, then the length of $$C G$$ is      ."},{"type":"image_path","image_path":"images/6412_q0.png"}],"answer":"2"}
{"id":"6419","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$A D = 10$$, E is a point on $$C D$$, and $$\\triangle A D E$$ is folded along line $$A E$$ such that point D lands exactly at point F on side $$B C$$. Then $$C E =$$      ."},{"type":"image_path","image_path":"images/6419_q0.png"}],"answer":"3"}
{"id":"6444","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the regular hexagon $$A B C D E F$$, $$M$$ and $$N$$ are two points on the diagonal $$B E$$. Add one of the following conditions: ① $$B M = E N$$; ② $$\\angle F A N = \\angle C D M$$; ③ $$A M = D N$$; ④ $$\\angle A M B = \\angle D N E$$. The conditions that can make quadrilateral $$A M D N$$ a parallelogram are          (fill in all the serial numbers of the conditions that meet the requirements)."},{"type":"image_path","image_path":"images/6444_q0.png"}],"answer":"①②④"}
{"id":"6451","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, point E lies on the diagonal AC of square ABCD, EF ⊥ AB at point F, connect DE and extend it to intersect side BC at point M and side AB extended at point G. If AF = 2, FB = 1, then MG =          ."},{"type":"image_path","image_path":"images/6451_q0.png"}],"answer":"$$\\frac{3 \\sqrt{5}}{2}$$"}
{"id":"6469","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt}\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = 5$$, $$A C = 10$$, $$D$$ is the midpoint of $$B C$$, $$E$$ is the midpoint of $$A C$$, connect $$B E$$ intersecting $$A D$$ at point $$F$$, then the area of $$\\triangle A B F$$ is     ."},{"type":"image_path","image_path":"images/6469_q0.png"}],"answer":"$$\\frac{25}{3}$$"}
{"id":"6478","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$ABCD$$, $$AB = 4$$, the angle bisector of $$\\angle BAD$$ intersects the extension of $$BC$$ at point E and intersects $$DC$$ at point F, and point F is the midpoint of side $$DC$$. $$DG \\bot AE$$, with foot at G. If $$DG = 1$$, then the length of $$AE$$ is          ."},{"type":"image_path","image_path":"images/6478_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"6486","difficulty":"0.6","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, $$\\angle A = 65^\\circ$$. Fold $$\\angle B$$ and $$\\angle C$$ as shown in the figure. If $$\\angle A D B' = 35^\\circ$$, then $$\\angle 1 + \\angle 2 + \\angle 3 =$$     $$^\\circ$$."},{"type":"image_path","image_path":"images/6486_q0.png"}],"answer":"$$265 \\circ$$"}
{"id":"6495","difficulty":"0.6","question_list":[{"type":"text","text":"Place a protractor and an unmarked transparent straightedge as shown in the figure, with the edges of the straightedge intersecting the protractor at points A, B, C, D, where points C and D correspond to the markings of 120 and 60 on the protractor. If the diameter EF of the protractor is 8 cm, then the distance from point O to CD is           cm."},{"type":"image_path","image_path":"images/6495_q0.png"}],"answer":"$$2 \\sqrt{3}$$"}
{"id":"6509","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, with diagonals $$A C$$ and $$B D$$ intersecting at point $$O$$, $$\\angle B A C = 90 \\circ$$, $$A H \\bot B D$$ at point $$H$$, $$A B = 2$$, $$B C = 2 \\sqrt{3}$$, then the length of $$A H$$ is        ."},{"type":"image_path","image_path":"images/6509_q0.png"}],"answer":"$$\\frac{2}{3} \\sqrt{3}$$/$$\\frac{2 \\sqrt{3}}{3}$$"}
{"id":"6513","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, fold the rectangular paper $$A B C D$$ so that $$A D$$ coincides with $$B C$$, obtaining the crease $$E F$$. Flatten the paper again, and fold it once more so that the image point $$A^{'}$$ of point A lies on $$E F$$, and the crease passes through point B, obtaining the crease $$B M$$. Connect $$M F$$. If $$M F \\bot B M$$ and $$A B = 6 \\text{cm}$$, then the length of $$A D$$ is            $$\\text{cm}$$."},{"type":"image_path","image_path":"images/6513_q0.png"}],"answer":"$$5 \\sqrt{3}$$"}
{"id":"6519","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 2, BC = 1. Rotate rectangle ABCD clockwise 90° about vertex C to obtain rectangle EFCG. Connect AE, take the midpoint H of AE, and connect DH. Then $$D H =$$       ."},{"type":"image_path","image_path":"images/6519_q0.png"}],"answer":"$$\\frac{\\sqrt{2}}{2}$$"}
{"id":"6544","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, chord $$CD \\bot AB$$ at point $$E$$, and $$AC = CD$$. If $$AC = 2\\sqrt{3}$$, then the radius of $$\\bigodot O$$ is      ."},{"type":"image_path","image_path":"images/6544_q0.png"}],"answer":"2"}
{"id":"6556","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A B = 10$$, $$B C = 8$$, $$A D$$ is the angle bisector of $$\\triangle A B C$$. $$M$$ is a moving point on side $$A C$$, and $$N$$ is a moving point on segment $$A D$$. Connect $$B M$$, $$C N$$, and $$M N$$. When $$C N + M N$$ is minimized, the area of $$\\triangle A B M$$ is     ."},{"type":"image_path","image_path":"images/6556_q0.png"}],"answer":"$$\\frac{72}{5}$$"}
{"id":"6560","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $$AC$$ and $$BD$$ of quadrilateral $$ABCD$$ intersect at point $$O$$, with $$OA = OB = OC = OD$$. Draw $$OE \\bot BD$$ intersecting $$BC$$ at point $$E$$. If $$AB = 5$$ and $$BE = 7$$, then the length of $$CE$$ is       ."},{"type":"image_path","image_path":"images/6560_q0.png"}],"answer":"$$2 \\sqrt{6}$$"}
{"id":"6564","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is inscribed in $$\\bigodot O$$ and $$\\angle A C B = 90 \\circ$$, chord $$C D$$ bisects $$\\angle A C B$$, and connect $$A D$$, $$B D$$. If $$A B = 5$$, $$A C = 4$$, then $$B D =$$     , $$C D =$$        ."},{"type":"image_path","image_path":"images/6564_q0.png"}],"answer":"$$\\frac{5}{2} \\sqrt{2}$$     $$\\frac{7}{2} \\sqrt{2}$$"}
{"id":"6588","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the acute triangle $$\\triangle ABC$$, $$AC = 7$$, the area of $$\\triangle ABC$$ is 21, and the angle bisector of $$\\angle BAC$$ intersects $$BC$$ at point D. M and N are moving points on $$AD$$ and $$AB$$, respectively. The minimum value of $$BM + MN$$ is        ."},{"type":"image_path","image_path":"images/6588_q0.png"}],"answer":"6"}
{"id":"6594","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, given $$AD = AE$$, please add one condition such that $$\\triangle ADC \\sim \\triangle AEB$$. The condition you add is     . (Do not add any letters or auxiliary lines)"},{"type":"image_path","image_path":"images/6594_q0.png"}],"answer":"$$A B = A C$$ or $$\\angle A D C = \\angle A E B$$ or $$\\angle A B E = \\angle A C D$$."}
{"id":"6619","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $$AF$$ and $$HD$$ of the regular octagon $$ABCDEFGH$$ intersect at point $$M$$. Then the measure of $$\\angle AMH$$ is        °."},{"type":"image_path","image_path":"images/6619_q0.png"}],"answer":"67.5"}
{"id":"6639","difficulty":"0.6","question_list":[{"type":"text","text":"In the isosceles right triangle $$\\triangle A B C$$, $$A C = B C$$, $$D$$ is a point on side $$A B$$, and $$\\frac{B D}{A D} = \\frac{1}{3}$$. Connect $$C D$$, and construct $$\\text{Rt} \\triangle D E C$$ downward with $$C D$$ as the hypotenuse. If $$\\angle E D C = 30^\\circ$$ and $$B C = 2$$, then the length of $$D E$$ is     ."},{"type":"image_path","image_path":"images/6639_q0.png"}],"answer":"$$\\frac{\\sqrt{30}}{4}$$"}
{"id":"6674","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, rotate $$\\triangle A B C$$ about point A to the position of $$\\triangle A E F$$, with point E on side $$B C$$, and $$E F$$ intersecting $$A C$$ at point G. If $$\\angle B = 70 \\circ$$, $$\\angle C = 25 \\circ$$, then $$\\angle F G C =$$        ."},{"type":"image_path","image_path":"images/6674_q0.png"}],"answer":"$$65 \\circ$$"}
{"id":"6697","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle A C B = 90 \\circ, A C = 8, B C = 6$$, point P is a moving point in the plane such that $$A P = 3$$, and Q is the midpoint of $$B P$$. During the motion of point P, let the length of segment $$C Q$$ be m. Then the range of m is          ."},{"type":"image_path","image_path":"images/6697_q0.png"}],"answer":"$$\\frac{7}{2}$$≤m≤$$\\frac{13}{2}$$"}
{"id":"6711","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = BC, ∠ABC = 90°, F is a point on the extension of AB, and point E is on BC such that AE = CF. If ∠BAE = 25°, then ∠ACF =          degrees."},{"type":"image_path","image_path":"images/6711_q0.png"}],"answer":"70"}
{"id":"6715","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point $$M$$ is a point on $$AB$$. From point $$D$$, draw $$DN \\bot DM$$, intersecting the extension of $$BC$$ at point $$N$$. Connect $$MN$$, and let point $$E$$ be the midpoint of $$MN$$. Connect $$BE$$. If $$AM = 1$$ and $$AD = 3$$, then the length of segment $$BE$$ is          ."},{"type":"image_path","image_path":"images/6715_q0.png"}],"answer":"$$\\sqrt{5}$$"}
{"id":"6721","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle B = 90 \\circ$$, $$A B = 2$$, $$C D = 8$$. Connect $$A C$$, $$A C \\bot C D$$, if $$s i n \\angle A C B = \\frac{1}{3}$$, then the length of $$A D$$ is         ."},{"type":"image_path","image_path":"images/6721_q0.png"}],"answer":"10"}
{"id":"6725","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the equilateral triangle $$\\triangle A B C$$ is inscribed in $$\\bigodot O$$. If the radius of $$\\bigodot O$$ is $$4$$, then the area of the shaded region is      ."},{"type":"image_path","image_path":"images/6725_q0.png"}],"answer":"$$\\frac{16 \\pi}{3} - 4 \\sqrt{3}$$"}
{"id":"6728","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$, $$AB = 6$$. With D, the midpoint of $$BC$$, as the center and $$BD$$ as the radius, draw an arc intersecting $$AB$$ and $$AC$$ at points E and F, respectively. Then, with point A as the center and $$AE$$ as the radius, draw another arc. The area of the shaded region in the figure is       ."},{"type":"image_path","image_path":"images/6728_q0.png"}],"answer":"$$6 \\pi - 9 \\sqrt{3}$$"}
{"id":"6743","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is inscribed in $$\\bigodot O$$, $$B D$$ is the diameter of $$\\bigodot O$$, $$A B = A C$$, $$\\angle A = 70 \\circ$$, then $$\\angle A B D - \\angle C B D =$$        $$\\circ$$."},{"type":"image_path","image_path":"images/6743_q0.png"}],"answer":"15"}
{"id":"6748","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, point E lies on $$A B$$ such that $$A E = 1$$, and P is a moving point on diagonal $$B D$$. The minimum value of the perimeter of $$\\triangle A E P$$ is       ."},{"type":"image_path","image_path":"images/6748_q0.png"}],"answer":"6"}
{"id":"6749","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 2, point E is the midpoint of $$B C$$, $$A E$$ intersects $$B D$$ at point P, F is a point on $$C D$$, connecting $$A F$$ intersects $$B D$$ and $$D E$$ at points M and N respectively, and $$A F \\bot D E$$, connecting $$P N$$,"},{"type":"image_path","image_path":"images/6749_q0.png"},{"type":"text","text":"then (I) the length of $$A E$$ is         .  \n(II) the length of $$P N$$ is         ."}],"answer":"$$\\sqrt{5}$$     $$\\frac{2 \\sqrt{65}}{15}$$"}
{"id":"6751","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, let $$G$$ be the centroid of $$\\Delta A B C$$. Draw a line through point $$G$$ parallel to $$B C$$, intersecting sides $$A B$$ and $$A C$$ at points $$D$$ and $$E$$, respectively. Let $$\\overset{\\rightarrow}{G B} = \\overset{\\rightarrow}{a}$$, $$\\overset{\\rightarrow}{G C} = \\overset{\\rightarrow}{b}$$. Express the vector $$\\overset{\\rightarrow}{D E}$$ in the form $$x \\overset{\\rightarrow}{a} + y \\overset{\\rightarrow}{b}$$ (where $$x$$ and $$y$$ are real numbers): _______________."},{"type":"image_path","image_path":"images/6751_q0.png"}],"answer":"$$- \\frac{2}{3} \\overset{\\rightarrow}{a} + \\frac{2}{3} \\overset{\\rightarrow}{b}$$"}
{"id":"6754","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$\\angle ACB = 90^\\circ$$, $$AC = 4$$, $$BC = 3$$. Rotate $$\\triangle ABC$$ about point $$B$$ to the position of $$\\triangle DBE$$, where point $$D$$ corresponds to point $$A$$, and point $$E$$ corresponds to point $$C$$. If the area of the shaded region in the figure is 4.5, then the tangent of $$\\angle CBE$$ is          ."},{"type":"image_path","image_path":"images/6754_q0.png"}],"answer":"$$\\frac{9}{13}$$"}
{"id":"6763","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, points $$D$$ and $$E$$ are the midpoints of sides $$A B$$ and $$B C$$, respectively. Point $$F$$ lies on the extension of segment $$D E$$, and $$\\angle B F C = 90 \\circ$$. If $$A C = 4$$ and $$B C = 8$$, then the length of $$D F$$ is      ."},{"type":"image_path","image_path":"images/6763_q0.png"}],"answer":"6"}
{"id":"6797","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$\\angle A = 30 \\circ$$, $$B C = 2$$, point $$D$$ is the midpoint of $$A C$$, point $$E$$ is a moving point on the hypotenuse $$A B$$, fold $$\\triangle A D E$$ along the line $$D E$$ to the position $$\\triangle A^{'} D E$$, $$A^{'} D$$ intersects $$A B$$ at point $$F$$, if $$\\triangle B A^{'} F$$ is a right triangle, then the length of $$A E$$ is        ."},{"type":"image_path","image_path":"images/6797_q0.png"}],"answer":"$$1$$ or $$\\frac{6}{5}$$"}
{"id":"6799","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A B = B C = \\frac{5 \\sqrt{2}}{2}$$, $$C D = 3$$, $$\\angle A B C = \\angle A D C = 90 \\circ$$, and point $$E$$ lies on $$B D$$ such that $$\\angle A E C = 135 \\circ$$. Then $$A E =$$      ."},{"type":"image_path","image_path":"images/6799_q0.png"}],"answer":"$$\\sqrt{10}$$"}
{"id":"6800","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$ with side length $$a$$, $$BF$$ is the median on side $$AC$$ and $$BF = b$$. Point $$D$$ lies on $$BF$$, and connect $$AD$$. Construct an equilateral triangle $$\\triangle ADE$$ on the right side of $$AD$$, and connect $$EF$$. The minimum perimeter of $$\\triangle AEF$$ is   , and at this time $$\\angle CFE =$$      ."},{"type":"image_path","image_path":"images/6800_q0.png"}],"answer":"$$\\frac{1}{2} a + b$$     $$90 \\circ$$"}
{"id":"6808","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of rhombus ABCD is 6, ∠ABC = 120°, M is a trisection point of side BC, and P is a moving point on diagonal AC. When PB + PM is minimized, the length of PM is ______."},{"type":"image_path","image_path":"images/6808_q0.png"}],"answer":"$$\\frac{\\sqrt{7}}{2}$$"}
{"id":"6815","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the rhombus $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, with $$A C = 8$$ and $$B D = 6$$. Points E and F lie on sides $$A B$$ and $$C D$$ respectively (point E does not coincide with A or B). Moreover, $$D E \\parallel B F$$, and $$D E$$ and $$B F$$ intersect $$A C$$ at points P and Q respectively. Connect $$B P$$ and $$D Q$$. The following four conclusions are given: ① $$A C$$ bisects the perimeter of quadrilateral $$B E D F$$; ② quadrilateral $$B E D F$$ is a rectangle; ③ $$B D$$ bisects $$\\angle P D Q$$; ④ when $$D E \\bot A B$$, $$\\frac{A E}{E D} = \\frac{7}{24}$$. Among the above conclusions, the serial numbers of all correct conclusions are           ."},{"type":"image_path","image_path":"images/6815_q0.png"}],"answer":"①③④"}
{"id":"6837","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A = \\alpha$$, the angle bisectors of $$\\angle A B C$$ and $$\\angle A C D$$ intersect at point $$A_{1}$$, yielding $$\\angle A_{1}$$; the angle bisectors of $$\\angle A_{1} B C$$ and $$\\angle A_{1} C D$$ intersect at point $$A_{2}$$, yielding $$A_{2}$$; $$\\hdots$$; the angle bisectors of $$\\angle A_{2019} B C$$ and $$\\angle A_{2019} C D$$ intersect at point $$A_{2020}$$, yielding $$\\angle A_{2020}$$, then $$\\angle A_{2020} =$$      ."},{"type":"image_path","image_path":"images/6837_q0.png"}],"answer":"$$\\frac{\\alpha}{2^{2020}}$$"}
{"id":"6849","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$ABC$$ with side length 6, point D lies on $$BC$$ such that $$BD = 4$$, and $$CQ$$ is the angle bisector of the exterior angle $$\\angle ACP$$ of $$\\triangle ABC$$. Reflecting $$\\triangle ABD$$ over $$AD$$ yields $$\\triangle AED$$, and $$DE$$ intersects $$CQ$$ at point F. Find the length of $$CF$$."},{"type":"image_path","image_path":"images/6849_q0.png"}],"answer":"$$\\frac{6}{5}$$"}
{"id":"6857","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the rhombus $$A B C D$$ has side length 4, $$\\angle A B C = 60 \\circ$$, points E and F are moving points on sides $$A D$$ and $$C D$$ respectively, and $$\\angle B E F = 120 \\circ$$. What is the maximum length of segment $$D F$$?"},{"type":"image_path","image_path":"images/6857_q0.png"}],"answer":"$$12 - 8 \\sqrt{2}$$"}
{"id":"6859","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, AB = 6, E is the midpoint of side CD, and F is a moving point on segment BC. Fold triangle ECF along the line EF to obtain triangle EC'F, and connect AC'. The minimum value of AC' is ______."},{"type":"image_path","image_path":"images/6859_q0.png"}],"answer":"$$3 \\sqrt{5} - 3$$/$$- 3 + 3 \\sqrt{5}$$"}
{"id":"6868","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, if point O is the midpoint of diagonal BD of rectangle ABCD, fold it as shown in the figure such that side AB falls onto BD and side CD also falls onto BD, with points A and C coinciding exactly at point O. Connect EC, intersecting BD at point G and DF at point H."},{"type":"image_path","image_path":"images/6868_q0.png"},{"type":"text","text":"(1) The measure of angle AEB is ______ degrees; (2) The value of GH/HC is ______."}],"answer":"$$60$$     $$\\frac{4}{5}$$"}
{"id":"6917","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, with the vertices $A$, $B$, and $C$ of the equilateral triangle $\\triangle ABC$ as centers, and with $AB$ as the radius, arcs are drawn. The closed figure formed by these three arcs is called a Reuleaux triangle. If the perimeter of the Reuleaux triangle is $2\\pi$, then the area of the Reuleaux triangle is          ."},{"type":"image_path","image_path":"images/6917_q0.png"}],"answer":"$$2 \\pi - 2 \\sqrt{3}$$"}
{"id":"6925","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, points E and F are on sides $$AD$$ and $$AB$$ respectively, and $$DE = BF$$. Connect $$BE$$ and $$CF$$. Then the minimum value of $$BE + CF$$ is           ."},{"type":"image_path","image_path":"images/6925_q0.png"}],"answer":"$$4 \\sqrt{5}$$"}
{"id":"6941","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A C B D$$, diagonals $$A B$$, $$C D$$ intersect at point O, $$\\angle A C B = 90 \\circ$$, $$B D = C D = 10$$, $$B C = 16$$, if $$\\angle D A B = 2 \\angle A B C$$, then the value of $$\\frac{A D}{A B}$$ is                   ."},{"type":"image_path","image_path":"images/6941_q0.png"}],"answer":"$$\\frac{1}{2}$$/0.5"}
{"id":"6944","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, point O is the midpoint of diagonal $$AC$$, point Q is a moving point on segment $$OA$$ (point Q does not coincide with point O or A). Connect $$BQ$$ and extend it to intersect side $$AD$$ at point E. From point Q, draw $$FQ \\bot BQ$$ intersecting $$CD$$ at point F. Connect $$BF$$ and $$EF$$. $$BF$$ intersects diagonal $$AC$$ at point G. From point C, draw $$CH \\parallel QF$$ intersecting $$BE$$ at point H. Connect $$AH$$. The following four conclusions: ① $$BQ = QF$$; ② The perimeter of $$\\triangle DEF$$ is 8; ③ $$\\angle BQG = \\angle BEF$$; ④ The minimum value of segment $$AH$$ is $$2\\sqrt{5} - 2$$. Among these, the correct conclusions are      . (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/6944_q0.png"}],"answer":"①②④"}
{"id":"6978","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle A C B = 54 \\circ, \\angle B A C = 64 \\circ$$, diagonal $$B D$$ bisects $$\\angle A B C$$, and $$\\angle B C D + \\angle D C A = 180 \\circ$$. Then $$\\angle A D C$$ is          degrees."},{"type":"image_path","image_path":"images/6978_q0.png"}],"answer":"59"}
{"id":"7056","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ intersects $$B D$$ at point O, and $$A C \\bot B D$$, $$A B = A C = B D$$. Point E is the intersection of the angle bisectors of $$\\angle A B D$$ and $$\\angle C A B$$. Connect $$E D$$ and $$E C$$. Then  \n(1) $$\\angle A E B =$$      ;  \n(2) Among the following conclusions: ① $$A D = D C$$; ② $$\\angle A C E = \\angle A B E$$; ③ $$B E \\bot E C$$; ④ $$S_{ \\triangle A E B} = S_{\\triangle E D C}$$; the correct ones are      ."},{"type":"image_path","image_path":"images/7056_q0.png"}],"answer":"$$135 \\circ$$     ②③④"}
{"id":"7061","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A = 30 \\circ$$, $$B C = 4$$, and $$D$$ is a moving point on $$A B$$. Construct an isosceles $$\\text{Rt} \\triangle D C E$$ with $$D C$$ as the hypotenuse to the right side such that $$\\angle C E D = 90 \\circ$$. Connect $$B E$$. What is the minimum value of segment $$B E$$?"},{"type":"image_path","image_path":"images/7061_q0.png"}],"answer":"$$\\sqrt{2}$$"}
{"id":"7064","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$BD$$ is the angle bisector of the isosceles $$\\triangle ABC$$, $$AB = AC = 6$$, $$BC = 8$$, $$E$$ is a moving point on segment $$BD$$ (excluding endpoints), connect $$AE$$, construct $$\\angle EAF = \\angle BAC$$, and $$AE = AF$$, connect $$DF$$. When the perimeter of $$\\triangle ADF$$ is minimized, the value of $$\\frac{AF}{DF}$$ is      ."},{"type":"image_path","image_path":"images/7064_q0.png"}],"answer":"$$\\frac{7}{4}$$."}
{"id":"7069","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C A B = 40 \\circ$$. Now fold $$\\triangle A B C$$ along a straight line passing through point $$A$$, such that point $$B$$ lands at point $$E$$ on the extension of segment $$A C$$. The angle bisector of $$\\angle A C B$$ intersects the fold line at point $$D$$. Connect $$B D$$ and $$D E$$. If one interior angle of $$\\triangle C D E$$ is three times another interior angle, then the measure of $$\\angle A C B$$ is      ."},{"type":"image_path","image_path":"images/7069_q0.png"}],"answer":"$$80 \\circ$$ or $$110 \\circ$$ or $$\\frac{500 \\circ}{3}$$."}
{"id":"7079","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, point E is the midpoint of side $$A B$$, and point P moves along side $$B C$$. Fold $$\\triangle B E P$$ along the crease $$E P$$ to obtain $$\\triangle F E P$$, and connect $$D F$$. If $$A B = 4$$, $$\\angle B = \\text{60} \\circ$$, then $$E F =$$   , and the minimum value of $$D F$$ is          ."},{"type":"image_path","image_path":"images/7079_q0.png"}],"answer":"2     $$2 \\sqrt{7} - 2$$"}
{"id":"7087","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, let $$P$$ be a point on side $$BC$$ of parallelogram $$ABCD$$. Fold $$\\triangle ABP$$ along line $$AP$$, such that point $$B$$ lands at point $$E$$ inside parallelogram $$ABCD$$, and $$EA = ED$$. If $$AB = 5$$, $$AD = 8$$, and the sine of $$\\angle B$$ is $$0.8$$, then the length of $$BP$$ is      ."},{"type":"image_path","image_path":"images/7087_q0.png"}],"answer":"$$\\frac{25}{7}$$"}
{"id":"7088","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$t a n A = \\frac{4}{3}$$, $$A D = \\frac{55}{4}$$, point E lies on side $$A D$$. Fold $$\\triangle A B E$$ along line $$B E$$ such that the image of side $$A B$$, denoted $$A^{'} B$$, is perpendicular to $$D C$$, with foot at F. $$A^{'} E$$ intersects $$D C$$ at point G. When point G is exactly the midpoint of $$A^{'} E$$, the length of $$A B$$ is          ."},{"type":"image_path","image_path":"images/7088_q0.png"}],"answer":"14"}
{"id":"7112","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C$$, $$\\angle B = 30 \\circ$$, $$B C = 9$$, $$D$$ is a point on $$A C$$ such that $$A D = 2 D C$$, and $$P$$ is a moving point on side $$B C$$. When $$\\triangle A P D$$ is a right triangle, the length of $$B P$$ is      ."},{"type":"image_path","image_path":"images/7112_q0.png"}],"answer":"3 or 6 or 7"}
{"id":"7132","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, and points $$E$$, $$F$$ lie on sides $$D C$$, $$B C$$ respectively, with $$B F = C E$$, and $$A E$$ bisects $$\\angle C A D$$. Connect $$D F$$, which intersects $$A E$$ and $$A C$$ at points $$G$$ and $$M$$ respectively. Point $$P$$ is a moving point on segment $$A G$$. From point $$P$$, draw $$P N \\bot A C$$, with foot of perpendicular at $$N$$. Connect $$P M$$. Then the minimum value of $$P M + P N$$ is , and $$S_{\\Delta A D M} =$$    ."},{"type":"image_path","image_path":"images/7132_q0.png"}],"answer":"$$2 \\sqrt{2}$$     $$4 \\sqrt{2}$$"}
{"id":"7144","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$\\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A B = 6 \\sqrt{2}$$, fold $$\\triangle A B C$$ along a certain straight line such that point $$B$$ lands on the midpoint of $$A C$$. If the crease intersects $$A B$$ at point $$M$$, then the length of $$A M$$ is      ."},{"type":"image_path","image_path":"images/7144_q0.png"}],"answer":"$$\\frac{7 \\sqrt{2}}{2}$$"}
{"id":"7152","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$A B = 2$$, $$B C = 4$$, point E lies on side BC, and segment AE is rotated clockwise about point E by $$90 \\circ$$ to obtain $$F E$$, and $$C F$$ is connected. The minimum value of $$C F$$ is      ."},{"type":"image_path","image_path":"images/7152_q0.png"}],"answer":"$$\\sqrt{2}$$"}
{"id":"7161","difficulty":"0.4","question_list":[{"type":"text","text":"Mathematician Fehrbe proposed the idea of using diagrams instead of calculations; such diagrams are called \"nomograms.\" The figure shown is a nomogram illustrating the relationship among x, y, and z. It consists of three rays a, b, and c emanating from point O, each with identical scales, and the endpoint of each ray is marked as 0. Rays a and c, as well as b and c, intersect at an angle of $$60 \\circ$$. Points A and B are selected on rays a and b, respectively, corresponding to scale values x and y. Connecting A and B with a straightedge intersects ray c at point C, and the scale value at point C is z.\n(1) If $$x = 20$$, $$y = 12$$, then z =        ;\n(2) If $$x = 2 y$$, then $$\\frac{z}{y} =$$        ."},{"type":"image_path","image_path":"images/7161_q0.png"}],"answer":"$$7.5$$     $$\\frac{2}{3}$$"}
{"id":"7202","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A C$$ is a diagonal, $$A E \\bot B C$$ at point $$E$$, point $$F$$ lies on the extension of $$A E$$ such that $$\\angle C A F = \\angle C F A$$, and ray $$C F$$ intersects the extension of segment $$A B$$ at point $$G$$. If $$A B = \\sqrt{10}$$, $$A D = 4$$, $$\\tan \\angle A B C = 3$$, then the length of $$B G$$ is      ; if point $$M$$ is the midpoint of $$A G$$, and segment $$M E$$ is connected, then the length of $$M E$$ is      ."},{"type":"image_path","image_path":"images/7202_q0.png"}],"answer":"$$2 \\sqrt{10}$$     $$\\frac{3 \\sqrt{2}}{2}$$"}
{"id":"7216","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, $$AB = 6$$, $$BC = 5$$, E and F are moving points on sides AB and BC respectively, and $$EF = 4$$, M is the midpoint of EF, and P is a moving point on side AD. Then the minimum value of $$CP + PM$$ is      ."},{"type":"image_path","image_path":"images/7216_q0.png"}],"answer":"11"}
{"id":"7237","difficulty":"0.4","question_list":[{"type":"text","text":"$$\\triangle A B C$$ is an equilateral triangle with side length 5, and $$\\triangle D C E$$ is an equilateral triangle with side length 3. The line BD intersects the line AE at point F. As shown in the figure, rotate $$\\triangle D C E$$ around point C by one full revolution. During this rotation, the minimum length of segment AF is           ."},{"type":"image_path","image_path":"images/7237_q0.png"}],"answer":"$$4 - \\sqrt{3}$$/$$- \\sqrt{3} + 4$$"}
{"id":"7248","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, and $$E$$ is a moving point on diagonal $$BD$$. An isosceles right triangle $$AEF$$ is constructed to the right with $$AE$$ as the hypotenuse. Then the minimum value of $$FC + FD$$ is      ."},{"type":"image_path","image_path":"images/7248_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"7255","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ and $$B D$$ intersect at point $$E$$, $$A C = B C$$, $$\\angle A C B = \\angle A D B = 90^\\circ$$, $$B D = a$$, $$A D = b$$, express the area of $$\\triangle B C D$$ in terms of $$a$$ and $$b$$ as      ."},{"type":"image_path","image_path":"images/7255_q0.png"}],"answer":"$$\\frac{a^{2} - a b}{4}$$"}
{"id":"7257","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point P lies on side $$AD$$, point E is symmetric to point B with respect to line $$CP$$, and ray $$ED$$ intersects the extension of $$CP$$ at point F. If $$AD = 4PD$$ and $$EF = 16\\sqrt{2}$$, then the length of $$BC$$ is      ."},{"type":"image_path","image_path":"images/7257_q0.png"}],"answer":"$$4 \\sqrt{17}$$"}
{"id":"7258","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$S_{\\triangle A B C} = 10$$, points D and E lie on $$B C$$ and $$A C$$ respectively, $$C D = 2 B D$$, $$C E = 2 A E$$, and $$B E$$ intersects $$A D$$ at point F. Then the area of $$\\triangle A F E$$ is        ."},{"type":"image_path","image_path":"images/7258_q0.png"}],"answer":"$$\\frac{4}{3}$$"}
{"id":"7270","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\triangle A B C$$, with $$B C = 6 \\text{cm}, \\angle A = 60 \\circ$$, the maximum value of $$A B + \\frac{\\sqrt{3} - 1}{2} A C$$ is        ."},{"type":"image_path","image_path":"images/7270_q0.png"}],"answer":"$$6 \\sqrt{2}$$"}
{"id":"7285","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AD = 4$$, there is a moving point C on side $$AD$$; construct equilateral $$\\triangle ABC$$ and equilateral $$\\triangle CDE$$ above side $$AD$$ with sides $$AC$$ and $$CD$$ respectively; connect $$BE$$, take the midpoint $$F$$ of side $$BE$$, and connect $$CF$$; then the minimum value of $$CF$$ is       ."},{"type":"image_path","image_path":"images/7285_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"7288","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point $$M$$ is the midpoint of $$B C$$, $$A D$$ bisects $$\\angle B A C$$, and $$B D \\bot A D$$ at point $$D$$. Extend $$B D$$ to intersect $$A C$$ at point $$N$$. If $$A B = 12$$, $$A C = 18$$, then $$M D =$$                       ."},{"type":"image_path","image_path":"images/7288_q0.png"}],"answer":"$$3$$"}
{"id":"7300","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = 3$$, $$\\angle B A C = 120^\\circ$$, construct $$\\angle A C M = \\angle A C B$$ on the other side of $$C A$$ with respect to $$\\angle A C B$$. Let point D be any point on side $$B C$$ (excluding endpoints). On ray $$C M$$, take $$C E = B D$$. Connect $$A D$$, $$D E$$, and $$A E$$. Let $$A C$$ intersect $$D E$$ at point F. Then the maximum value of segment $$C F$$ is        ."},{"type":"image_path","image_path":"images/7300_q0.png"}],"answer":"$$\\frac{9}{4}$$"}
{"id":"7306","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, AB∥CD, AC∥BD, CE bisects ∠ACD and intersects BD at point E. Point F lies on the extension of CD, and ∠BEF = ∠CEF. If ∠DEF = ∠EDF, then the measure of ∠A is     $$\\circ$$."},{"type":"image_path","image_path":"images/7306_q0.png"}],"answer":"108"}
{"id":"7319","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$\\triangle ABC$$, $$\\angle BAC = 150^\\circ$$, point $$D$$ lies on $$AB$$ such that $$AD = 1$$ and $$BD = 6$$, and point $$E$$ lies on side $$BC$$. If the point $$F$$, which is the image of point $$E$$ rotated counterclockwise by $$15^\\circ$$ about point $$D$$, lies exactly on $$AC$$, then the length of $$BE$$ is            ."},{"type":"image_path","image_path":"images/7319_q0.png"}],"answer":"$$1 + 6 \\sqrt{2}$$"}
{"id":"7348","difficulty":"0.4","question_list":[{"type":"text","text":"In quadrilateral $$A B C D$$, $$E$$, $$F$$ are the midpoints of sides $$A B$$, $$A D$$ respectively. If $$B C = 15$$, $$C D = 9$$, $$E F = 6$$, $$\\angle A F E = 55^\\circ$$, then $$\\angle A D C =$$      ."},{"type":"image_path","image_path":"images/7348_q0.png"}],"answer":"145°"}
{"id":"7355","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, point $$D$$ is a moving point on side $$A C$$, connect $$B D$$, rotate segment $$B D$$ clockwise around point $$B$$ by $$60 \\circ$$ to obtain segment $$B E$$, connect $$C E$$. If $$A B = 2 \\sqrt{10}$$, $$B C = 3 A B$$, then the minimum value of segment $$C E$$ is      ."},{"type":"image_path","image_path":"images/7355_q0.png"}],"answer":"$$9 \\sqrt{3} - 3$$"}
{"id":"7358","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$ and $$\\triangle A D E$$, $$\\angle B A C = \\angle D A E = 90 \\circ$$, $$A B = A C = 2 A D = 2 A E = 4$$, point O lies on side $$B C$$ such that $$O C = 3 O B$$. Rotate $$\\triangle A D E$$ clockwise about point A, connect $$C E$$, and let P be the midpoint of $$C E$$. Then the maximum value of $$O P$$ is          ."},{"type":"image_path","image_path":"images/7358_q0.png"}],"answer":"$$\\sqrt{10} + 1$$"}
{"id":"7365","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point P is inside ∠AOB, and points M and N are moving points on sides OA and OB, respectively, and points M and N do not coincide with point O."},{"type":"image_path","image_path":"images/7365_q0.png"},{"type":"text","text":"(1) If point P is moved to a position inside ∠AOB such that OP bisects ∠AOB, and PN ∥ OA, ON = 2, then the length of PN is ______; (2) If ∠AOB = 60°, OP = a, and as points M and N move, when the perimeter of △PMN is minimized, the distance from point O to line MN is ______. (Expressed as an algebraic expression in terms of a)"}],"answer":"2     $$\\frac{1}{2} a$$"}
{"id":"7369","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$ABCD$$ with side length 4, $$\\angle ABC = 120^\\circ$$. Translate $$\\triangle ADC$$ along the ray $$AC$$ to obtain $$\\triangle A'D'C'$$. Connect $$A'B$$ and $$D'B$$. The minimum value of $$A'B + D'B$$ is      ."},{"type":"image_path","image_path":"images/7369_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"7386","difficulty":"0.4","question_list":[{"type":"text","text":"Given: As shown in the figure, the area of the equilateral triangle $$A B C$$ is $$3 \\sqrt{3}$$, and $$D$$, $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, with $$A D = C E$$. Then the minimum value of $$D E$$ is      ."},{"type":"image_path","image_path":"images/7386_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"7394","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 8$$, points E and F lie on sides $$A D$$ and $$B C$$ respectively, and $$A E = 3$$. Folding along line $$E F$$, the image of point A, denoted $$A^{'}$$, falls exactly on diagonal $$A C$$, and the image of point B is $$B^{'}$$. Point M is a moving point on segment $$A A^{'}$$. Find the minimum value of $$E M + \\frac{\\sqrt{5}}{5} A^{'} M$$        ."},{"type":"image_path","image_path":"images/7394_q0.png"}],"answer":"$$\\frac{12}{5}$$/$$2 . 4$$"}
{"id":"7403","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C = 2$$, point $$D$$ is a moving point on side $$B C$$ (not coinciding with points $$B$$ or $$C$$), $$C E$$ is perpendicular to $$A D$$, intersecting $$A B$$ at point $$E$$, with foot of perpendicular at point $$H$$, connect $$B H$$ and extend it to intersect $$A C$$ at point $$F$$,\n① If $$A D$$ is the median of side $$B C$$, then $$D H = \\frac{\\sqrt{5}}{5}$$;\n② If $$A D$$ bisects $$\\angle C A B$$, then $$\\frac{C D}{B D} = \\frac{\\sqrt{2}}{2}$$;\n③ If $$B D = 2 C D$$, then $$A E = 2 B E$$;\n④ The minimum value of $$B H$$ is $$\\sqrt{5} - 1$$.\nThe correct sequence numbers above are      ."},{"type":"image_path","image_path":"images/7403_q0.png"}],"answer":"①②④"}
{"id":"7411","difficulty":"0.4","question_list":[{"type":"text","text":"In quadrilateral $$A B D C$$, $$A C = 3$$, $$A B = 5$$, $$B D = C D$$, $$\\angle B D C =\\text{90} \\circ$$, then the maximum value of $$A D$$ is      ."},{"type":"image_path","image_path":"images/7411_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"7425","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 8$$, and there is a moving point P on diagonal $$AC$$ (point P does not coincide with point C or point A). Construct square $$DPFG$$ with $$DP$$ as a side. If E is the midpoint of $$DC$$, connect $$EG$$, then the minimum value of $$EG$$ is          ."},{"type":"image_path","image_path":"images/7425_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"7434","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the equilateral triangle $$\\triangle A B C$$ has side length $$4$$, and $$D$$, $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, with $$B D = A E$$. Let $$F$$ be the midpoint of $$D E$$, and connect $$A F$$. When $$A F = \\frac{\\sqrt{13}}{2}$$, the length of $$B D$$ is     ."},{"type":"image_path","image_path":"images/7434_q0.png"}],"answer":"$$1$$"}
{"id":"7442","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, points D and E lie on $$A C$$ and $$B C$$ respectively, and $$\\angle C D E = \\angle B$$. When $$\\triangle C D E$$ is folded along $$D E$$, point C lands exactly at point F on side $$A B$$. If $$A C = 8$$ and $$B C = 6$$, then the length of $$C D$$ is"},{"type":"image_path","image_path":"images/7442_q0.png"}],"answer":"$$\\frac{25}{8}$$"}
{"id":"7467","difficulty":"0.4","question_list":[{"type":"text","text":"In rhombus ABCD, ∠D = 60°, CD = 4, E is a point inside the rhombus such that AE = 2. Connect CE, let F be the midpoint of CE, connect BF, let G be the midpoint of BF, connect AG. Then the maximum value of AG is     ."},{"type":"image_path","image_path":"images/7467_q0.png"}],"answer":"$$\\frac{1}{2} + \\sqrt{7}$$"}
{"id":"7470","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4, A D = 6$$, points E and F lie on sides $$A B$$ and $$C D$$ respectively, point M is a moving point on segment $$E F$$, and from point M, a perpendicular to $$E F$$ intersects sides $$A D$$ and $$B C$$ at points G and H respectively. If segment $$E F$$ exactly bisects the area of rectangle $$A B C D$$, and $$D F = 1$$, then the length of $$G H$$ is      ."},{"type":"image_path","image_path":"images/7470_q0.png"}],"answer":"$$\\frac{4}{3} \\sqrt{10}$$"}
{"id":"7510","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ and quadrilateral $$C E F G$$ are both squares. Point E is a moving point on the extension of $$D C$$, and point G lies on ray $$C B$$ (not coinciding with point C). H is the midpoint of $$D F$$, and connect $$G H$$. If $$A D = 8$$, then the minimum value of $$G H$$ is       ."},{"type":"image_path","image_path":"images/7510_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"7513","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A B = 6$$, $$A D = 2$$, the area of quadrilateral $$A B C D$$ is $$10 \\sqrt{2}$$, connect diagonals $$A C$$, $$B D$$, where $$B D \\bot A D$$, then the minimum value of $$A C + B C$$ is       ."},{"type":"image_path","image_path":"images/7513_q0.png"}],"answer":"$$4 \\sqrt{6}$$"}
{"id":"7517","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle with side length $$5$$, point $$D$$ is a point outside $$\\triangle A B C$$, $$\\angle B A D > \\angle B A C$$, $$A D = A C$$. If $$C D = 6$$, connect $$B D$$, then the length of segment $$B D$$ is           ."},{"type":"image_path","image_path":"images/7517_q0.png"}],"answer":"$$4 + 3 \\sqrt{3}$$"}
{"id":"7532","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠ACB = 90°, AC = 4, BC = 3, point D is the midpoint of AC. Rotate CD counterclockwise around point C; during the rotation, the corresponding point of D is point E. Connect AE and BE. The minimum area of triangle AEB is       ."},{"type":"image_path","image_path":"images/7532_q0.png"}],"answer":"1"}
{"id":"7538","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point $$P$$ is a moving point on the diagonal $$BD$$ of square $$ABCD$$, $$PE \\bot BC$$ at point $$E$$, $$PF \\bot CD$$ at point $$F$$, and connect $$EF$$. The following 5 conclusions are given: ① $$AP = EF$$; ② $$AP \\bot EF$$; ③ $$\\triangle APD$$ is always an isosceles triangle; ④ $$\\angle PFE = \\angle BAP$$; ⑤ The minimum value of $$EF$$ equals $$\\frac{1}{2} BD$$. The correct conclusion(s) is/are      ."},{"type":"image_path","image_path":"images/7538_q0.png"}],"answer":"①②④⑤"}
{"id":"7543","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C$$, $$\\angle A C B = 90 \\circ$$, $$A E$$ bisects $$\\angle B A C$$ and intersects $$B C$$ at point $$E$$, $$B D \\bot A E$$ intersects the extension of $$A E$$ at point $$D$$, $$D M \\bot A C$$ intersects the extension of $$A C$$ at point $$M$$, connect $$C D$$. The following conclusions: ① $$\\angle C D A = 45 \\circ$$; ② $$C D = \\frac{1}{2} A E$$; ③ $$A C + C E = A D$$; ④ $$\\frac{A C + A B}{A M}$$ is a constant. The correct conclusions are      . (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/7543_q0.png"}],"answer":"①②④"}
{"id":"7553","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle A B C$$, $$A B = 4$$, $$A D \\parallel B C$$, $$A D = \\frac{1}{2} A B$$, point $$E$$ is a moving point on $$A C$$, connect $$D E$$, point $$F$$ is the midpoint of $$D E$$, connect $$B F$$, then the minimum value of $$B F$$ is      ."},{"type":"image_path","image_path":"images/7553_q0.png"}],"answer":"$$\\frac{5 \\sqrt{3}}{2}$$/$$\\frac{5}{2} \\sqrt{3}$$"}
{"id":"7561","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 3 \\sqrt{3}$$, $$P$$ is a point on $$B C$$, and an equilateral triangle $$\\triangle A P Q$$ is constructed with $$A P$$ as a side (with points $$A$$, $$P$$, $$Q$$ arranged counterclockwise). Connect $$C Q$$ and $$D Q$$. Then the minimum value of $$C Q + D Q$$ is      ."},{"type":"image_path","image_path":"images/7561_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"7563","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point E is a point on side $$AD$$, and point F lies on the extension of side $$DC$$, with $$AE = CF$$. Connect $$EF$$ and let it intersect diagonal $$AC$$ at point G. If $$AB = 8$$ and $$AE = 2$$, then the length of segment $$DG$$ is      ."},{"type":"image_path","image_path":"images/7563_q0.png"}],"answer":"$$\\sqrt{34}$$"}
{"id":"7611","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$B A = B C = 4$$, point D is the midpoint of side $$B C$$. Connect $$A D$$, draw $$B E \\bot A D$$ through point B, intersecting at point E, and extend $$B E$$ to intersect $$A C$$ at point F. Then the length of $$E F$$ is          ."},{"type":"image_path","image_path":"images/7611_q0.png"}],"answer":"$$\\frac{8 \\sqrt{5}}{15}$$/$$\\frac{8}{15} \\sqrt{5}$$"}
{"id":"7628","difficulty":"0.4","question_list":[{"type":"text","text":"In the isosceles triangle $$A B C$$, $$A C = B C = 4$$, $$\\angle A C B = 90 \\circ$$, $$D$$ is any point in the plane, connect $$A D$$, when $$A D = 2$$, rotate $$A D$$ clockwise 90° about point $$A$$ to obtain $$A E$$, connect $$D E, B D, B E$$, take the midpoint $$M$$ of $$B D$$, connect $$C M$$. When $$B, M, E$$ are collinear, the length of $$C M$$ is           ."},{"type":"image_path","image_path":"images/7628_q0.png"}],"answer":"$$\\frac{\\sqrt{30} + \\sqrt{2}}{2}$$ or $$\\frac{\\sqrt{30} - \\sqrt{2}}{2}$$."}
{"id":"7634","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$B C = 4$$, $$\\angle A B C = 60^{\\circ}$$, on side $$B C$$ there is a segment $$E F$$ moving from $$B$$ to $$C$$, stopping when point $$F$$ reaches point $$C$$, with point $$E$$ to the left of point $$F$$, $$E F = 1$$, connect $$A E$$ and $$A F$$, then the minimum perimeter of $$\\triangle A E F$$ is      ."},{"type":"image_path","image_path":"images/7634_q0.png"}],"answer":"$$8$$"}
{"id":"7640","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, point E lies on $$A D$$, $$B C = 3 E D$$, connect $$B E$$, $$D F \\bot A B$$ at point F, intersecting $$B E$$ at point G, $$\\angle A B E = 2 \\angle A D F$$, $$B G = G D$$, if $$C D = 5$$, then the length of segment $$B F$$ is           ."},{"type":"image_path","image_path":"images/7640_q0.png"}],"answer":"2"}
{"id":"7677","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, quadrilateral $$E B C F$$ is folded along $$E F$$, causing point $$B$$ and point $$C$$ to land at points $$B^{'}$$ and $$C^{'}$$ respectively. Then, $$\\triangle A E F$$ is folded along $$A F$$, causing point $$E$$ to land at point $$E^{'}$$. If $$\\angle B A C = 50 \\circ$$ and $$\\angle 1 = 65 \\circ$$, then the measure of $$\\angle 3$$ is      ."},{"type":"image_path","image_path":"images/7677_q0.png"}],"answer":"$$85 \\circ$$"}
{"id":"7686","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A = 90 \\circ$$, $$A B = A C$$, $$A D = 4$$, $$A B = 10$$, points $$D$$, $$E$$ lie on sides $$A B$$, $$A C$$ respectively, $$A D = A E$$, connect $$D C$$, points $$M$$, $$P$$, $$N$$ are the midpoints of $$D E$$, $$D C$$, $$B C$$ respectively."},{"type":"image_path","image_path":"images/7686_q0.png"},{"type":"text","text":"(1) The area of $$\\triangle P M N$$ is     , (2) When $$\\triangle A D E$$ is rotated freely about point $$A$$ in the plane, the maximum area of $$\\triangle P M N$$ is        ."}],"answer":"$$\\frac{9}{2}$$     $$\\frac{49}{2}$$"}
{"id":"7696","difficulty":"0.4","question_list":[{"type":"text","text":"Through studying the textbook section \"13.4 Shortest Path Problem\", we have experienced the role of axial symmetry transformations. Please use your knowledge of axial symmetry to solve the following problem: As shown in the figure, $$C$$ is the midpoint of $$AB$$, $$\\angle ACD + \\angle BCE = 60^\\circ$$, $$AD = 2$$, $$BE = 4.5$$, $$AB = 6$$, then the maximum value of $$DE$$ is      ."},{"type":"image_path","image_path":"images/7696_q0.png"}],"answer":"9.5"}
{"id":"7697","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in a set of triangle rulers, the longer leg of the triangle containing the $$30^\\circ$$ angle ($$\\triangle ABC$$) coincides with the hypotenuse of the triangle containing the $$45^\\circ$$ angle ($$\\triangle ACD$$). Points P and Q lie on sides $$AC$$ and $$BC$$, respectively, $$AB$$ intersects $$CD$$ at E, and quadrilateral $$EPQB$$ is a parallelogram with area 3. Find the length of segment $$DE$$."},{"type":"image_path","image_path":"images/7697_q0.png"}],"answer":"$$\\frac{3 \\sqrt{2} - \\sqrt{6}}{2}$$"}
{"id":"7729","difficulty":"0.4","question_list":[{"type":"text","text":"If a rhombus has one diagonal equal to its side length, it is called a \"perfect rhombus.\" As shown in the figure, given that the \"perfect rhombus\" ABCD has side length 4, BD is its shorter diagonal, and points M and N are two moving points on sides AD and CD respectively, satisfying AM + CN = 4. Let the area of triangle BMN be S. Then the range of S is     ."},{"type":"image_path","image_path":"images/7729_q0.png"}],"answer":"$$3 \\sqrt{3}$$≤S≤$$4 \\sqrt{3}$$"}
{"id":"7732","difficulty":"0.4","question_list":[{"type":"text","text":"(1) If the polynomial $$m x^{3} + n x^{2} - 61 x - 36$$ is divisible by $$2 x + 1$$ and $$3 x - 4$$, then $$m - n =$$_________.\n(2) As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = \\angle B C A = 44 \\circ$$, $$M$$ is a point inside $$\\triangle A B C$$ such that $$\\angle M C A = 30 \\circ$$, $$\\angle M A C = 16 \\circ$$, then the measure of $$\\angle B M C$$ is_________."},{"type":"image_path","image_path":"images/7732_q0.png"}],"answer":"(1)$$- 10$$;(2)$$150 \\circ$$"}
{"id":"7740","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, and $$C$$ is a moving point on $$\\bigodot O$$. Construct a square $$ACEF$$ on the left side of $$AC$$. Connect $$BF$$. Then the maximum value of $$\\frac{BF}{AB}$$ is         ."},{"type":"image_path","image_path":"images/7740_q0.png"}],"answer":"$$\\frac{\\sqrt{5} + 1}{2}$$"}
{"id":"7758","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A B = 3$$, $$B C = 8$$, $$\\angle A B C = 60 \\circ$$, $$F$$ is the midpoint of $$B C$$, and $$E$$ is a point on the extension of $$C D$$. If $$A F$$ bisects $$\\angle B A E$$, then $$D E =$$       ."},{"type":"image_path","image_path":"images/7758_q0.png"}],"answer":"7"}
{"id":"7777","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point E lies on $$AB$$, with $$AE : BE = 2 : 1$$. Connect $$DE$$, and from point A draw $$AG \\bot DE$$ intersecting $$DE$$ at point $$G$$. Extend $$AG$$ to intersect $$BC$$ at point $$F$$. Let the ratio of the area of quadrilateral $$BEGF$$ to the area of $$\\triangle AEG$$ be $$a$$, and the ratio of the area of square $$ABCD$$ to the area of $$\\triangle AEG$$ be $$b$$. Then $$a + b =$$       ."},{"type":"image_path","image_path":"images/7777_q0.png"}],"answer":"12"}
{"id":"7789","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, this is a rectangular football field, with $$AB$$ as the goal, $$CD \\bot AB$$ at point D, and $$AB = a$$ meters. A player dribbles along $$CD$$ toward the goal $$AB$$ and prepares to shoot from point Q. It is known that $$BD = 3a$$ meters and $$QD = 3a$$ meters. When the goalkeeper spreads his arms, the defensive range is approximately $$0.25a$$ meters. At this moment, the goalkeeper stands within the angle $$\\angle AQB$$, with arms spread as $$MN$$ perpendicular to $$AQ$$ for defense. When the midpoint of $$MN$$ is at a distance of _____ meters from $$AB$$, the goalkeeper can just successfully defend."},{"type":"image_path","image_path":"images/7789_q0.png"}],"answer":"$$\\frac{37}{20} a$$/$$\\frac{37 a}{20}$$"}
{"id":"7791","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B = 90 \\circ$$, point $$E$$ lies on $$A B$$, connect $$C E$$, fold $$\\triangle B C E$$ along the straight line $$C E$$ to obtain $$\\triangle F C E$$, and point $$F$$, the image of point $$B$$, lies exactly on $$A C$$. Draw $$F D \\bot A B$$ with foot at point $$D$$, point $$M$$ is a point on the extension of $$D F$$, connect $$C M$$. Point $$N$$ lies on $$C M$$, point $$R$$ lies on $$C F$$, take a point $$Q$$ on the extension of $$C M$$, connect $$F N$$, $$R N$$, and $$R Q$$. If $$\\angle C F N = \\angle Q$$, $$R N = F R = C R$$, $$F N = N Q$$, $$A B = \\frac{40}{3}$$, $$A F = \\frac{1}{2} A B$$, then the length of segment $$R Q$$ is      ."},{"type":"image_path","image_path":"images/7791_q0.png"}],"answer":"$$5 \\sqrt{2} + 5$$"}
{"id":"7796","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = 2 A D = 4$$, $$\\angle D = 60^\\circ$$, point $$P$$ is a point on side $$C D$$, connect $$P B$$, fold $$\\triangle B C P$$ along $$P B$$ so that point $$C$$ lands at point $$N$$, where $$C P \\geq 2$$, let $$P N$$ intersect $$A B$$ at point $$M$$, if the area of $$\\triangle B M P$$ is $$x$$, then the range of $$x$$ is      ."},{"type":"image_path","image_path":"images/7796_q0.png"}],"answer":"$$\\sqrt{3} \\leq x \\leq \\frac{7 \\sqrt{3}}{5}$$"}
{"id":"7802","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 8, point $$E$$ is a point on side $$B C$$ such that $$B E = 2$$, point $$F$$ is the midpoint of side $$A B$$, connect $$E F$$, and construct an isosceles right triangle $$\\text{Rt} \\triangle E G F$$ with $$E F$$ as one leg to the right, such that $$\\angle E F G = 90^\\circ$$, connect $$C G$$, then the length of $$C G$$ is      ."},{"type":"image_path","image_path":"images/7802_q0.png"}],"answer":"$$2 \\sqrt{13}$$"}
{"id":"7819","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of semicircle $$O$$, chord $$AC$$ is parallel to radius $$OD$$, connect $$CD$$, $$CD = AO$$, rotate semicircle $$O$$ about point $$A$$, with $$O$$'s corresponding point being $$O^{'}$$, semicircle $$O^{'}$$ intersects $$AB$$ at points $$E$$ and $$D$$, when points $$E$$, $$D$$, and $$B^{'}$$ lie on the same straight line, if $$BB^{'} = 8 \\sqrt{2}$$, then the area of quadrilateral $$AODC$$ is      ."},{"type":"image_path","image_path":"images/7819_q0.png"}],"answer":"$$32 \\sqrt{3}$$"}
{"id":"7837","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$AC = 10$$, $$BC = 14$$, $$\\tan \\angle ACB = \\frac{4}{3}$$, rotate $$\\triangle ABC$$ about point C to obtain $$\\triangle DEC$$, with points A and B corresponding to points D and E respectively, connect $$AD$$. If points M and N are the midpoints of $$BC$$ and $$AD$$ respectively, connect $$MN$$, then the range of possible lengths of $$MN$$ is           ."},{"type":"image_path","image_path":"images/7837_q0.png"}],"answer":"$$4 \\sqrt{2} - 5 \\leq M N \\leq 4 \\sqrt{2} + 5$$"}
{"id":"7855","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, there is a right triangle paper $$ABC$$ containing a $$30^\\circ$$ angle, with the shortest side $$BC = 1$$. Three rhombuses of maximum area are folded out, each using $$\\angle A$$, $$\\angle B$$, and $$\\angle C$$ as one of their interior angles. The sum of the maximum and minimum areas of these three largest rhombuses is      ."},{"type":"image_path","image_path":"images/7855_q0.png"}],"answer":"$$42 - \\frac{214 \\sqrt{3}}{9}$$"}
{"id":"7861","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rectangles $$A B C D$$ and $$A E F G$$, with $$A D = 12$$, $$A B = 9$$, $$A G = 8$$, $$A E = 6$$. Rectangle $$A E F G$$ rotates about point A. The following conclusions are given: ① $$4 B E = 3 D G$$; ② $$B E \\bot D G$$; ③ $$D E^{2} + B G^{2} = 315$$; ④ When $$\\angle B A G = 60 \\circ$$, $$4 S_{\\Delta A B G} = 3 \\sqrt{3} S_{\\Delta A D G}$$. Among these, the correct conclusions are            ."},{"type":"image_path","image_path":"images/7861_q0.png"}],"answer":"①②④"}
{"id":"7862","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = A C = 25, B C = 30$$, point $$D$$ is a moving point on side $$A C$$, and $$\\triangle B C D$$ is folded along $$B D$$ to obtain $$\\triangle B E D$$, with $$B E$$ intersecting $$A C$$ at point $$F$$. Then the maximum value of $$E F$$ is            ."},{"type":"image_path","image_path":"images/7862_q0.png"}],"answer":"6"}
{"id":"7876","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC (AB > BC), G lies on the extension of CB. The perpendicular bisector of side AC, DE, intersects the angle bisector of ∠ABG at point M, intersects AB at point D, and intersects AC at point E. MN ⊥ AB at N. Given AB = 13, BC = 9, MN = 3, the area of △BMN is    ."},{"type":"image_path","image_path":"images/7876_q0.png"}],"answer":"3"}
{"id":"7878","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AD$$ is the altitude of $$\\triangle ABC$$, $$\\angle BAC = 45^\\circ$$. If $$AD = 18$$ and $$DC = 6$$, then the area of $$\\triangle ABC$$ is      ."},{"type":"image_path","image_path":"images/7878_q0.png"}],"answer":"$$135$$"}
{"id":"7900","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4. Extend $$C B$$ to $$E$$ such that $$E B = 2$$, and construct square $$E F G B$$ above with $$E B$$ as a side. Extend $$F G$$ to intersect $$D C$$ at $$M$$. Connect $$A M$$ and $$A F$$. Let $$H$$ be the midpoint of $$A D$$, and connect $$F H$$, which intersects $$A B$$ and $$A M$$ at points $$N$$ and $$K$$, respectively. The following conclusions are given: ① $$\\triangle A N H \\sim \\triangle G N F$$; ② $$F K = 3 N K$$; ③ $$\\angle A F N = \\angle H F G$$; ④ $$S_{\\triangle A F N} : S_{\\triangle A D M} = 1 : 4$$. Among these, the correct conclusions are     ."},{"type":"image_path","image_path":"images/7900_q0.png"}],"answer":"①②④"}
{"id":"7907","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, fold the square ABCD with side length 2 along point A, such that point B lands at $$B^{'}$$, connect D$$B^{'}$$, and let point F be the midpoint of D$$B^{'}$$. Then the minimum value of CF is      ."},{"type":"image_path","image_path":"images/7907_q0.png"}],"answer":"$$\\sqrt{5}$$-1/-1+$$\\sqrt{5}$$"}
{"id":"7922","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$A D = 4$$, point $$E$$ is the midpoint of $$A D$$, point $$P$$ is a moving point on $$B E$$, point $$Q$$ is the midpoint of $$P C$$, connect $$A Q$$, then the minimum length of $$A Q$$ is         ."},{"type":"image_path","image_path":"images/7922_q0.png"}],"answer":"$$\\frac{12 \\sqrt{13}}{13}$$"}
{"id":"7967","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 3$$, $$B C = 4$$, points D and E are the midpoints of sides $$B C$$ and $$A C$$ respectively, connect $$D E$$, point F is a moving point on side $$A B$$, and $$C F = D E$$, then the length of $$A F$$ is      ."},{"type":"image_path","image_path":"images/7967_q0.png"}],"answer":"$$2.5$$ or $$1.1$$"}
{"id":"7971","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 16, BC = 12, E is the midpoint of side BC, point F lies on side AB, and ∠EDF = 45°. Find the length of AF."},{"type":"image_path","image_path":"images/7971_q0.png"}],"answer":"$$\\frac{60}{11}$$"}
{"id":"7979","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$\\angle A = 30^\\circ$$. A sufficiently large right triangle ruler $$PMN$$ ($$\\angle M = 90^\\circ$$, $$\\angle MPN = 30^\\circ$$) is placed as shown in the figure, with vertex P sliding along side AC, and the leg $$PM$$ of the triangle ruler always passing through point B, while the hypotenuse $$PN$$ intersects $$AB$$ at point D. If point P slides such that both $$\\triangle PAD$$ and $$\\triangle PBC$$ are isosceles triangles, then the measure of $$\\angle C$$ is        ."},{"type":"image_path","image_path":"images/7979_q0.png"}],"answer":"$$30 \\circ$$ or $$75 \\circ$$ or $$52.5 \\circ$$"}
{"id":"7984","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given $$\\angle M O N = 30 \\circ$$, points $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, … lie on ray ON, and points $$B_{1}$$, $$B_{2}$$, $$B_{3}$$, … lie on ray OM. $$\\Delta A_{1} B_{1} A_{2}$$, $$\\Delta A_{2} B_{2} A_{3}$$, $$\\Delta A_{3} B_{3} A_{4}$$, … are all equilateral triangles. If $$O A_{1} = 2$$, then the side length of $$\\Delta A_{5} B_{5} A_{6}$$ is        ."},{"type":"image_path","image_path":"images/7984_q0.png"}],"answer":"32"}
{"id":"7995","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, $$M$$ and $$N$$ are two points on $$\\overset{⌢}{AB}$$ (distinct from $$A$$ and $$B$$), $$C$$ is a moving point on $$\\overset{⌢}{MN}$$, the angle bisector of $$\\angle ACB$$ intersects $$\\bigodot O$$ at point $$D$$, and the angle bisector of $$\\angle BAC$$ intersects $$CD$$ at point $$E$$. When point $$C$$ moves from point $$M$$ to point $$N$$, the ratio of the lengths of the paths traced by points $$C$$ and $$E$$ is           ."},{"type":"image_path","image_path":"images/7995_q0.png"}],"answer":"$$\\sqrt{2}$$"}
{"id":"8000","difficulty":"0.4","question_list":[{"type":"text","text":"Square $$ABCD$$ has side length 8, and points $$E$$, $$F$$ lie on sides $$AD$$, $$BC$$ respectively. Quadrilateral $$ABFE$$ is folded along $$EF$$, such that point $$A$$ lands at $$A'$$ and point $$B$$ lands at point $$B'$$, and $$A'B'$$ intersects $$BC$$ at $$G$$. The following conclusions: ① When $$A'$$ is the midpoint of $$CD$$, the side lengths of $$\\triangle A'DE$$ are in the ratio $$3:4:5$$; ② Connect $$AA'$$, then $$AA' = EF$$; ③ When the side lengths of $$\\triangle A'DE$$ are in the ratio $$3:4:5$$, $$A'$$ is the midpoint of $$CD$$; ④ As $$A'$$ moves along $$CD$$, the perimeter of $$\\triangle A'CG$$ remains constant. Among these, the correct ones are           (write all correct conclusion numbers)."},{"type":"image_path","image_path":"images/8000_q0.png"}],"answer":"①②④"}
{"id":"8025","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$A B = 3$$, $$A C = 4$$, $$\\angle B A C = 90 \\circ$$, $$D$$ and $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, and $$B D = A E$$. Find the minimum value of $$C D + B E$$ ."},{"type":"image_path","image_path":"images/8025_q0.png"}],"answer":"$$\\sqrt{58}$$"}
{"id":"8027","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, points $$E$$ and $$F$$ are the midpoints of $$B C$$ and $$C D$$, respectively, $$\\angle E A F = 60^\\circ$$, $$A E = 3$$, $$A F = 6$$, then the length of $$A D$$ is      ."},{"type":"image_path","image_path":"images/8027_q0.png"}],"answer":"$$2 \\sqrt{13}$$"}
{"id":"8029","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rhombus ABCD, $$AE \\bot BC$$, point E is the foot of the perpendicular, point F is the midpoint of AE, connect BF and extend it to intersect AD at point G, connect CG, $$CE = 2, CG = 2\\sqrt{11}$$, then the length of AF is          ."},{"type":"image_path","image_path":"images/8029_q0.png"}],"answer":"$$\\sqrt{7}$$"}
{"id":"8030","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point $$P$$ is a moving point on side $$BC$$ of rectangle $$ABCD$$. Triangle $$PAB$$ is folded along line $$AP$$, and point $$B$$ lands at position $$B^{'}$$. Given: $$AB = 6$$, $$BC = 4$$, find the length of $$BP$$ when point $$B^{'}$$ lies exactly on the axis of symmetry of the rectangle."},{"type":"image_path","image_path":"images/8030_q0.png"}],"answer":"$$2 \\sqrt{3}$$ or $$18 - 12 \\sqrt{2}$$"}
{"id":"8034","difficulty":"0.4","question_list":[{"type":"text","text":"In future studies, we will learn this theorem: in a right triangle, if one acute angle equals 30°, then the side opposite this angle equals half of the hypotenuse. That is, as shown, in Rt△ABC, ∠ACB=90°, if ∠ABC=30°, then $$A C = \\frac{1}{2} A B$$. Problem: in Rt△ABC, ∠ACB=90°, ∠ABC=30°, AC=$$\\sqrt{3}$$, point D is the midpoint of side BC, point E is a moving point on hypotenuse AB, connect DE, fold △BDE along line DE so that point B maps to point F. When line DF⊥AB, the length of AE is      ."},{"type":"image_path","image_path":"images/8034_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{2}$$ or $$\\frac{\\sqrt{3}}{2}$$/$$\\frac{\\sqrt{3}}{2}$$ or $$\\frac{3 \\sqrt{3}}{2}$$"}
{"id":"8046","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$\\angle B = \\angle C = 90 \\circ$$, point $$E$$ is a point on side $$B C$$, $$\\triangle A D E$$ is an equilateral triangle, and $$\\frac{A B}{C D} = \\frac{n}{m}$$, then $$\\frac{B E}{C E} =$$      ."},{"type":"image_path","image_path":"images/8046_q0.png"}],"answer":"$$\\frac{2 m - n}{2 n - m}$$"}
{"id":"8049","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 8$$, the diameter of semicircle O is $$B C$$. Point E starts from D and moves toward C at a speed of 1 unit per second. Point F starts from B and moves toward A at a speed of 2 units per second. When point F reaches point A, point E also stops moving. Let the time of motion be $$t$$ seconds.  \n(1) When $$E F$$ is tangent to semicircle O, $$t =$$  \n(2) Point M is the midpoint of $$E F$$, and point N is the circumcenter of $$\\triangle M B C$$. The length of the path traced by point N is           ."},{"type":"image_path","image_path":"images/8049_q0.png"}],"answer":"2     $$\\frac{59}{36}$$"}
{"id":"8050","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, let AB be the diameter of semicircle O, and chords AC and BD intersect at point E. Point F lies on AC such that ∠DFE = ∠B. Given DF = 1 and cos ∠AED = √2/3, find the length of segment AB."},{"type":"image_path","image_path":"images/8050_q0.png"}],"answer":"$$\\frac{3 \\sqrt{2}}{2}$$"}
{"id":"8055","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, points E and F lie on sides AB and BC of rectangle ABCD, respectively. Connect EF. Fold triangle BEF along line EF to obtain triangle HEF. AB = 8, BC = 6, AE : EB = 3 : 1. Connect AH and HC. When point F moves along segment BC, the minimum area of quadrilateral AHCD is      ."},{"type":"image_path","image_path":"images/8055_q0.png"}],"answer":"$$32$$"}
{"id":"8061","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the central angle of sector $$OAB$$ is $$\\angle AOB = 45^\\circ$$, and $$OB = 2$$. The sector $$OAB$$ is translated along ray $$OA$$ to obtain sector $$O'A'B'$$, where arc $$\\overset{⌢}{AB}$$ intersects $$O'B'$$ at point C. If point C is the trisection point of arc $$\\overset{⌢}{AB}$$ closer to point A, then the area of the shaded region is              ."},{"type":"image_path","image_path":"images/8061_q0.png"}],"answer":"$$\\frac{\\pi}{3} + \\frac{\\sqrt{3} - 1}{2}$$"}
{"id":"8068","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given that the side length of square $$A B C D$$ is $$6$$, $$E$$ is a point on side $$A B$$ such that $$B E = 4$$, and $$F$$ is a point on side $$B C$$. Fold $$\\triangle E B F$$ along $$E F$$ so that the image of point $$B$$ is $$B^{'}$$. Connect $$A C$$, $$D F$$, $$D E$$, and $$B^{'} D$$. The following conclusions are given: ① If $$E F \\parallel A C$$, then $$D F = 2 \\sqrt{10}$$; ② If $$E F \\parallel A C$$, then $$B^{'} D = 2 \\sqrt{2}$$; ③ The maximum area of $$\\triangle D E F$$ is $$18$$; ④ The minimum value of $$B^{'} D$$ is $$2 \\sqrt{10} - 4$$. Among these, the correct ones are     ."},{"type":"image_path","image_path":"images/8068_q0.png"}],"answer":"①②③④"}
{"id":"8076","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A D = 2 A B = 4$$. $$E$$ is the midpoint of $$B C$$, and $$F$$ is a moving point on segment $$A B$$ (excluding endpoints). Connect $$A E, D E, D F, E F$$. Let $$A E$$ and $$D F$$ intersect at point $$G$$, and let $$M$$ be the midpoint of $$D G$$. Connect $$M E$$ and $$M C$$. Among the following conclusions: ① $$M C$$ is the perpendicular bisector of $$D E$$; ② Points $$A, D, E, F$$ may be concyclic; ③ The minimum perimeter of $$\\triangle M E F$$ is $$2 \\sqrt{10}$$; ④ If $$F$$ is the midpoint of $$A B$$, then $$\\triangle M E G$$ is an equilateral triangle. The correct ones are      . (Write all correct conclusion numbers)"},{"type":"image_path","image_path":"images/8076_q0.png"}],"answer":"①③"}
{"id":"8083","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point E is a moving point on side BC of square ABCD with side length 10. Connect DE, rotate segment DE counterclockwise 90° about point E to obtain segment EF, connect AF and DF. DF intersects AB at point G, and connect EG. When AF + DF is minimized, the length of segment EG is        ."},{"type":"image_path","image_path":"images/8083_q0.png"}],"answer":"$$\\frac{25}{3}$$/$$8 \\frac{1}{3}$$"}
{"id":"8099","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\Delta A B C$$, $$A B = 10$$, $$B D = 4$$, $$B E = 2$$. Point $$P$$ starts from point $$E$$ and moves along the direction of $$E A$$. Connect $$P D$$, and construct an equilateral triangle $$\\Delta D P F$$ on the right side of $$P D$$ in the manner shown in the figure. When point $$P$$ moves from point $$E$$ to point $$A$$, the length of the path traced by point $$F$$ is  ."},{"type":"image_path","image_path":"images/8099_q0.png"}],"answer":"8"}
{"id":"8109","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the right triangle $$B E F$$ has vertex $$F$$ moving along the diagonal $$A C$$ of rectangle $$A B C D$$. Connect $$A E$$. $$\\angle E B F = \\angle A C D$$, $$A B = 6$$, $$B C = 8$$, then the minimum value of $$A E$$ is     ."},{"type":"image_path","image_path":"images/8109_q0.png"}],"answer":"$$\\frac{72}{25}$$"}
{"id":"8118","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, △ABC is an isosceles triangle with AB = AC and ∠B = 30°. △ADE is a right triangle with ∠ADE = 90° and ∠E = 30°, and AD = AB. Rotate △ADE around point A so that AD and AE intersect BC at points F and G, respectively. When ∠AGB = 75°, $$\\frac{F G}{D E} =$$     ."},{"type":"image_path","image_path":"images/8118_q0.png"}],"answer":"$$\\frac{\\sqrt{3} - 1}{2}$$"}
{"id":"8120","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, segment $$AD$$ intersects segment $$BC$$ at point $$E$$, and connect $$AB$$, $$CD$$. If $$\\angle AEB = 60^{\\circ}$$, $$AD = 2$$, $$BC = 3$$, then the minimum value of $$AB + CD$$ is        ."},{"type":"image_path","image_path":"images/8120_q0.png"}],"answer":"$$\\sqrt{7}$$"}
{"id":"8132","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given square $$A B C D$$, points P and Q move along $$B D$$ and $$C D$$ respectively. Connect $$A P$$, $$A Q$$, and $$P Q$$, where $$A Q$$ intersects $$B D$$ at point E, and $$\\angle P A Q = 45^\\circ$$. Reflect $$\\triangle P Q E$$ over $$P Q$$ to obtain $$\\triangle P Q E^{'}$$. When point $$E^{'}$$ lies on $$B C$$, $$B P = \\sqrt{2}$$, then $$C Q =$$  , $$C E^{'} =$$      ."},{"type":"image_path","image_path":"images/8132_q0.png"}],"answer":"2     $$\\sqrt{5} - 1$$/$$- 1 + \\sqrt{5}$$"}
{"id":"8158","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that $$BE$$ and $$CE$$ bisect $$\\angle ABC$$ and $$\\angle ACD$$ respectively, and $$\\angle BEC = 30$$ degrees. Then $$\\angle EAC =$$      degrees."},{"type":"image_path","image_path":"images/8158_q0.png"}],"answer":"60"}
{"id":"8161","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\triangle$$ABC, $$\\angle$$ACB = 90°, AB = 2AC, AC = $$\\sqrt{2}$$, point E is a point on side AB. Fold $$\\triangle$$BCE along CE to obtain $$\\triangle$$B'CE. Draw AF parallel to BC, intersecting the angle bisector of $$\\angle$$ABC at point F. Connect $$B'F$$. The minimum length of $$B'F$$ is"},{"type":"image_path","image_path":"images/8161_q0.png"}],"answer":"$$\\sqrt{10} - \\sqrt{6}$$"}
{"id":"8164","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, two isosceles right triangles $$\\triangle A D B$$ and $$\\triangle A E C$$ are constructed on the same side with $$A B$$ and $$A C$$ as their hypotenuses, respectively. If point $$D$$ is the centroid of $$\\triangle A E C$$, then $$t a n \\angle B A C =$$      ."},{"type":"image_path","image_path":"images/8164_q0.png"}],"answer":"$$\\frac{1}{2}$$/$$0 . 5$$"}
{"id":"8191","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points $$E$$, $$F$$ are moving points on sides $$B C$$, $$C D$$ respectively. Connect $$A E$$, $$A F$$, and fold the rectangle along $$A E$$, $$A F$$ such that the corresponding sides of $$A B$$, $$A D$$, namely $$A B^{'}$$, $$A D^{'}$$, lie on the same straight line. If point $$F$$ is the midpoint of $$C D$$, then $$A E =$$         ."},{"type":"image_path","image_path":"images/8191_q0.png"}],"answer":"$$2 \\sqrt{5}$$"}
{"id":"8215","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\angle A B C$$ is a steel frame structure, with at most 5 equal-length steel bars that can be constructed inside the angle, satisfying $$B D = D E = E F = F G = G H = H I$$. Let $$\\angle A B C = x$$, then the range of values for x is      ."},{"type":"image_path","image_path":"images/8215_q0.png"}],"answer":"$$15 \\circ \\leq x < 18 \\circ$$"}
{"id":"8228","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B C A = 90 \\circ$$, $$A B = 2 A C$$, point P is inside $$\\triangle A B C$$, and $$P A = \\sqrt{3}$$, $$P B = 5$$, $$P C = 2$$. Find the area of $$\\triangle A B C$$   ."},{"type":"image_path","image_path":"images/8228_q0.png"}],"answer":"$$\\frac{6 + 7 \\sqrt{3}}{2}$$"}
{"id":"8230","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, point D is the midpoint of $$A B$$, connect $$D C$$, and draw $$D E \\bot D C$$ intersecting $$A C$$ at point E. If $$A B = 10$$, $$C E = 6$$, then the length of $$A E$$ is       ."},{"type":"image_path","image_path":"images/8230_q0.png"}],"answer":"$$\\frac{7}{3}$$/$$2 \\frac{1}{3}$$"}
{"id":"8233","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠B = 30°, ∠ACB = 90°, AB = 2, D is on BC. Rotate segment AD counterclockwise 60° about point A to obtain AP. Then the minimum value of CP is     ."},{"type":"image_path","image_path":"images/8233_q0.png"}],"answer":"$$\\frac{1}{2}$$"}
{"id":"8238","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\triangle ABC$$, $$\\angle ACB = 90^\\circ$$, $D$ is a point on $AB$, $AD = AC$, $E$ is a point on $AD$, through $E$ draw $EF \\bot CD$, intersecting $CD$ at point $F$ and intersecting $BC$ at point $G$, $CG = BD$, $BC = 7$, $DE = 2$, then $AC = $      ."},{"type":"image_path","image_path":"images/8238_q0.png"}],"answer":"$$\\frac{9}{2}$$"}
{"id":"8260","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, E is the midpoint of CD, connect BE, F is the midpoint of BE, connect AF. If $$AB = 2$$, $$BC = 5$$, $$\\angle BAD = 120^\\circ$$, then the length of AF is        ."},{"type":"image_path","image_path":"images/8260_q0.png"}],"answer":"$$\\frac{\\sqrt{\\text{19}}}{\\text{2}}$$"}
{"id":"8272","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle △ABC with side length 2, D is the midpoint of BC, and E is a point on side AC. The minimum value of BE + DE is     ."},{"type":"image_path","image_path":"images/8272_q0.png"}],"answer":"$$\\sqrt{7}$$."}
{"id":"8287","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles triangle $$\\triangle ABC$$, $$AB = AC$$, $$\\angle BAC = 45^\\circ$$, $$AD \\bot BC$$ at $$D$$. From point $$C$$, draw $$CE \\bot AB$$ at $$E$$, intersecting $$AD$$ at $$H$$; draw $$EF \\bot AC$$ at $$F$$, intersecting $$AD$$ at $$G$$; connect $$BH$$. The following conclusions: ① $$S_{\\triangle AEH} : S_{\\triangle ACH} = AE : AC$$; ② $$EG \\parallel BH$$; ③ $$AH = 2CH$$; ④ $$\\frac{1}{2} AD = EF$$; ⑤ $$AC = CE + GE$$. Among these, the correct ones are  ."},{"type":"image_path","image_path":"images/8287_q0.png"}],"answer":"①②⑤"}
{"id":"8290","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that rectangle $$A B C D$$ has two sides $$A B = 6$$, $$A D = 8$$, point $$E$$ is the intersection of diagonals $$A C$$ and $$B D$$, and point $$P$$ is a moving point on side $$A D$$. Construct the reflection of point $$D$$ over line $$P E$$, denoted as $$D^{'}$$. When $$E D^{'}$$ is perpendicular to one side of the rectangle, the length of $$P D$$ is      ."},{"type":"image_path","image_path":"images/8290_q0.png"}],"answer":"$$\\frac{5}{2}$$ or 5"}
{"id":"8302","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C = 6$$, point P is a moving point inside $$\\triangle A B C$$ such that the area of $$\\triangle A C P$$ is 3, and point Q is a moving point on $$A B$$. Find the minimum value of $$P B + P Q$$            ."},{"type":"image_path","image_path":"images/8302_q0.png"}],"answer":"$$5 \\sqrt{2}$$"}
{"id":"8324","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the area of $$\\triangle A D C$$ is 4, $$A D$$ bisects $$\\angle B A C$$, and $$A D \\bot B D$$ at point $$D$$. What is the area of $$\\triangle A B C$$?"},{"type":"image_path","image_path":"images/8324_q0.png"}],"answer":"8"}
{"id":"8339","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = A C$$, draw $$C D \\bot B C$$, connect $$D A$$, $$D B$$, draw $$A E \\bot B D$$ through point $$A$$, meeting at point $$E$$, if $$\\angle E A D = 2 \\angle A D C$$, and the area of $$\\triangle A D C$$ is 6, then the length of $$B C$$ is     ."},{"type":"image_path","image_path":"images/8339_q0.png"}],"answer":"$$4 \\sqrt{3}$$"}
{"id":"8347","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$ with side length 30, point M is a moving point on segment $$AB$$. The equilateral triangle $$\\triangle ABC$$ is folded along a line passing through M, with the crease intersecting line $$AC$$ at point N, such that point A lands at point D on line $$BC$$, and $$BD : DC = 1 : 4$$. Let the crease be $$MN$$, then the value of $$AN$$ is      ."},{"type":"image_path","image_path":"images/8347_q0.png"}],"answer":"21 or 65"}
{"id":"8352","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, ∠BAD = 90°, ∠ADC = 90°, ∠BCD = 60°, BC = CD. P is any point on the sides of quadrilateral ABCD. When AB = 4 and ∠APB = 30°, the length of BP is          ."},{"type":"image_path","image_path":"images/8352_q0.png"}],"answer":"8 or $$4 \\sqrt{3}$$ or 4"}
{"id":"8357","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 4$$, $$A C = 3$$, points $$D$$ and $$E$$ lie on $$A B$$ and $$A C$$ respectively, with $$B D = C E$$; $$M$$ and $$N$$ are the midpoints of $$B E$$ and $$C D$$ respectively; line $$M N$$ intersects $$A B$$ and $$A C$$ at points $$P$$ and $$Q$$; if $$P D = 1$$, then $$Q E =$$      ."},{"type":"image_path","image_path":"images/8357_q0.png"}],"answer":"2"}
{"id":"8359","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rectangle $$A B C D$$, with side $$A B = 2$$, $$B C = 2 \\sqrt{3}$$, fold the rectangle $$A B C D$$ so that point B lands on ray $$B D$$, and denote the corresponding point of B as $$B^{'}$$; the crease intersects sides $$A D$$ and $$B C$$ at points E and F respectively. When $$B^{'} D = 1$$, the length of $$A E$$ is       ."},{"type":"image_path","image_path":"images/8359_q0.png"}],"answer":"$$\\frac{\\sqrt{3}}{3}$$ or $$\\sqrt{3}$$"}
{"id":"8385","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, points $$M$$ and $$N$$ are moving points on sides $$BC$$ and $$CD$$ respectively (excluding endpoints), and $$\\angle MAN = 45^\\circ$$. The following four conclusions: ① When $$MN = \\sqrt{2} MC$$, then $$\\angle BAM = 22.5^\\circ$$; ② $$\\angle AMN + \\angle MNC = 90^\\circ$$; ③ The perimeter of $$\\triangle MNC$$ remains constant; ④ If $$DN = 2$$ and $$AB = 6$$, then the area of $$\\triangle ABM$$ is 9. The correct conclusion(s) is/are           ."},{"type":"image_path","image_path":"images/8385_q0.png"}],"answer":"①③④"}
{"id":"8506","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, points $$E$$ and $$F$$ lie on $$AC$$ such that $$\\frac{AE}{AC} = \\frac{1}{3}$$ and $$\\frac{CF}{AC} = \\frac{1}{4}$$. Extend $$DE$$ to intersect $$AB$$ at point $$G$$, and extend $$DF$$ to intersect $$BC$$ at point $$H$$. Connect $$GH$$. The following conclusions: ① Point $$G$$ is the midpoint of $$AB$$, ② $$DF = GH$$, ③ $$\\angle GDH = 45^\\circ$$, ④ $$DE \\cdot DG = DF \\cdot DH$$. The correct conclusions are ______. (Write all correct conclusion numbers)"},{"type":"image_path","image_path":"images/8506_q0.png"}],"answer":"①③④"}
{"id":"8544","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, O is the intersection point of the perpendicular bisectors of the three sides of triangle $$A B C$$. If $$\\angle A = 60^\\circ$$, $$A B = 4$$, $$A C = 3$$, and segment $$O B$$ is connected, then the length of $$O B$$ is        ."},{"type":"image_path","image_path":"images/8544_q0.png"}],"answer":"$$\\frac{\\sqrt{39}}{3}$$"}
{"id":"8545","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A C = 4$$, $$B C = 3$$, point $$D$$ is the midpoint of $$A C$$, and point $$E$$ is a moving point on side $$A B$$. Connect $$D E$$, and fold $$\\triangle A D E$$ along the line $$D E$$ to obtain $$\\triangle A^{'} D E$$. When $$A^{'} E \\parallel B C$$, the length of $$A E$$ is      ."},{"type":"image_path","image_path":"images/8545_q0.png"}],"answer":"1 or 4"}
{"id":"8554","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 4, and $$\\triangle B E F$$ is an equilateral triangle, with point F located above side $$B C$$, and point E moving along ray $$B C$$. Connect $$C F$$, and let M be the midpoint of $$C F$$. Then the minimum length of segment $$A M$$ is      ."},{"type":"image_path","image_path":"images/8554_q0.png"}],"answer":"$$2 + \\sqrt{3}$$/$$\\sqrt{3} + 2$$"}
{"id":"8556","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 10$$, point P is a moving point inside square $$ABCD$$. Connect $$AP$$, rotate $$AP$$ counterclockwise around point A by $$90^\\circ$$ to obtain segment $$AQ$$, connect $$QD$$ and $$BP$$, extend $$BP$$ to intersect line $$QD$$ at point M. When point P is the midpoint of $$BM$$, the minimum value of segment $$PC$$ is      ."},{"type":"image_path","image_path":"images/8556_q0.png"}],"answer":"$$\\frac{5 \\sqrt{10} - 5 \\sqrt{2}}{2}$$"}
{"id":"8558","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an equilateral triangle, $$\\triangle A B D$$ is an isosceles right triangle, $$\\angle B A D = 90 \\circ$$, $$A E \\bot B D$$ at point $$E$$, connect $$C D$$ intersecting $$A E$$ and $$A B$$ at points $$F$$ and $$G$$ respectively, draw $$A H \\bot C D$$ from point $$A$$ intersecting $$C D$$ and $$B D$$ at points $$P$$ and $$H$$ respectively, then the correct conclusion(s) is/are        .\n①$$\\angle B A C = 4 \\angle A D C$$; ②$$D F = A H$$; ③$$B H = E F$$; ④$$\\angle D A P = \\angle C G B$$"},{"type":"image_path","image_path":"images/8558_q0.png"}],"answer":"①②④"}
{"id":"8582","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in Rt△ABC, ∠C=90°, AC=8, BC=6. Rotate △ABC about point C to obtain △A'B'C, where A' is the image of point A. Let P be the midpoint of A'B', and connect BP. During the rotation, the maximum length of segment BP is      ."},{"type":"image_path","image_path":"images/8582_q0.png"}],"answer":"11"}
{"id":"8585","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, points $$E$$ and $$F$$ are on sides $$A B$$ and $$B C$$ respectively, $$B E = 4$$, $$C D = 8$$, $$\\angle F E D = 30 \\circ$$, $$\\angle F D E = 45 \\circ$$, then the length of $$B C$$ is      ."},{"type":"image_path","image_path":"images/8585_q0.png"}],"answer":"$$8 \\sqrt{3} + 4$$"}
{"id":"8586","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A B = 2$$, $$A D = 4$$, $$\\angle A = 60 \\circ$$, $$E$$ is the midpoint of $$A D$$, $$F$$ is a moving point on $$E C$$, $$G$$ is the midpoint of $$B F$$, connect $$G D$$, then the minimum value of $$G D$$ is"},{"type":"image_path","image_path":"images/8586_q0.png"}],"answer":"$$2$$"}
{"id":"8596","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, points D and E are two points inside $$\\triangle$$ABC, and DE$$/ /$$AB. Connect AD, BE, and CE. If AB=9$$\\sqrt{2}$$, DE=2$$\\sqrt{2}$$, BC=10, ∠ABC=75°, then the minimum value of AD+BE+CE is           ."},{"type":"image_path","image_path":"images/8596_q0.png"}],"answer":"$$13 \\sqrt{2}$$"}
{"id":"8617","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C$$, $$\\angle A C B = 90 \\circ$$, point D is the midpoint of AB, point E lies on AC, point F lies on BC, $$\\angle E D F = 90 \\circ$$, connect BE. If $$\\angle C B E = 2 \\angle E D A$$, $$C E = 6$$, then $$B E =$$          ."},{"type":"image_path","image_path":"images/8617_q0.png"}],"answer":"10"}
{"id":"8620","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A D \\bot B C$$ at point D, E is the midpoint of side $$A C$$, connect $$B E$$ intersecting $$A D$$ at F, fold $$\\triangle A F E$$ along $$A C$$ to $$\\triangle A G E$$, such that $$G E \\parallel A D$$. The following conclusions: ① Quadrilateral $$A F E G$$ is a rhombus; ② $$2 A E^{2} = C D \\cdot B C$$; ③ $$S_{\\triangle A B F} = S_{\\triangle C B F}$$; ④ Connect $$B G$$ intersecting $$A D$$ at H, then $$B H = H G$$. Among the above conclusions, the correct ones are     . (Fill in the correct serial numbers)."},{"type":"image_path","image_path":"images/8620_q0.png"}],"answer":"①③④"}
{"id":"8627","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point A is a point outside the equilateral triangle BCD. Connect AB and AD such that AB = AD. From point A, draw AE ∥ CD, intersecting BC and BD at points E and F, respectively. If 3BD = 5AE and EF = 6, then the length of segment AE is      ."},{"type":"image_path","image_path":"images/8627_q0.png"}],"answer":"9"}
{"id":"8634","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in sector $$A O B$$, $$\\angle A O B = 90^{\\circ}$$, radius $$O A = 6$$, point $$F$$ is located at $$\\frac{1}{3}$$ of arc $$A B$$ closer to point $$A$$, points $$C$$ and $$D$$ lie on segments $$O A$$ and $$O B$$ respectively, $$C D = 6$$, $$E$$ is the midpoint of $$C D$$, connect $$E F$$ and $$B E$$. As $$C D$$ slides (with $$C D$$ length remaining constant), when $$E F$$ takes its minimum value, the perimeter of the shaded region is          ."},{"type":"image_path","image_path":"images/8634_q0.png"}],"answer":"$$3 + 3 \\sqrt{3} + 2 \\pi$$"}
{"id":"8635","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, points $$E$$ and $$N$$ are the midpoints of $$B C$$ and $$A B$$, respectively. $$C N$$ intersects $$D E$$ at point $$G$$. Connect $$B G$$ and extend it to intersect $$C D$$ at point $$F$$. $$C N$$ intersects diagonal $$B D$$ at point $$H$$. The following conclusions are given: ① $$\\angle C N B + \\angle B E G = 180^\\circ$$; ② $$S_{A B C D} = 6 S_{\\triangle B N H}$$; ③ $$\\frac{N G}{C D} = \\frac{E G}{C F}$$; ④ $$N G + E G = \\sqrt{2} B G$$. Among these, the correct conclusions are     . (Fill in all the serial numbers of the correct conclusions)"},{"type":"image_path","image_path":"images/8635_q0.png"}],"answer":"①③④"}
{"id":"8641","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure: In △ABC, AB = AC, ∠BAC = 90°, the vertex P of the right angle ∠EPF is the midpoint of side BC, and the two sides PE and PF intersect AB and AC at points E and F, respectively. The following four conclusions are given:"},{"type":"image_path","image_path":"images/8641_q0.png"},{"type":"text","text":"① AE = CF; ② EF = AP; ③ 2S_{quadrilateral AEPF} = S_{△ABC}; ④ When ∠EPF rotates around vertex P within △ABC (point E does not coincide with A or B), BE + CF = EF. Among the above conclusions, the always correct ones are          ."}],"answer":"①③"}
{"id":"8650","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in equilateral triangle △ABC, BF is the median on side AC, and point D is a moving point on BF. Connect AD, and construct equilateral triangle △ADE on the right side of AD. Connect EF. When the perimeter of △AEF is minimized, the measure of ∠CFE is     ."},{"type":"image_path","image_path":"images/8650_q0.png"}],"answer":"$$90^{\\circ}$$"}
{"id":"8659","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle ABC$$, $$AB = AC$$, $$\\angle B = 40^\\circ$$, point $$D$$ is a moving point on $$BC$$, and $$\\triangle ABD$$ is folded along $$AD$$ to obtain $$\\triangle ADE$$. When the overlapping part of $$\\triangle ADE$$ and $$\\triangle ABC$$ is a right triangle, the measure of $$\\angle BAD$$ is                 ."},{"type":"image_path","image_path":"images/8659_q0.png"}],"answer":"$$25 \\circ$$ or $$50 \\circ$$ or $$75 \\circ$$"}
{"id":"8816","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, point $$E$$ is a point on side $$C D$$, and $$B C = E C$$. $$C F \\bot B E$$ intersects $$A B$$ at point $$F$$ and intersects $$B E$$ at point $$G$$. Point $$P$$ is a point on the extension of $$E B$$. The following conclusions are given: ① $$B E$$ bisects $$\\angle C B F$$; ② $$C F$$ bisects $$\\angle D C B$$; ③ $$B C = P B$$; ④ $$P F = P C$$. Among these, the correct ones are            . (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/8816_q0.png"}],"answer":"①②④"}
{"id":"8817","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, points $$E$$, $$F$$ lie on sides $$B C$$, $$C D$$ respectively, and $$\\angle E A F = 45^\\circ$$. $$A E$$ intersects $$B D$$ at point $$M$$, $$A F$$ intersects $$B D$$ at point $$N$$. The extension of $$F M$$ intersects the extension of $$C B$$ at point $$P$$, and $$B P = D F$$. Connect $$E F$$."},{"type":"image_path","image_path":"images/8817_q0.png"},{"type":"text","text":"(1) $$\\angle A M F =$$        .  \n(2) If $$D F = 2$$, $$E F = 5$$, then $$t a n \\angle C F E =$$        ."}],"answer":"$$90 \\circ$$     $$\\frac{3}{4}$$"}
{"id":"8835","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the diameter AB of semicircle O is 15, chord BC is 9, and chord BD bisects ∠ABC. Then the length of BD is          ."},{"type":"image_path","image_path":"images/8835_q0.png"}],"answer":"$$6 \\sqrt{5}$$"}
{"id":"8843","difficulty":"0.4","question_list":[{"type":"text","text":"Given $$\\triangle A B C$$, $$\\angle C = 90 \\circ$$, point $$E$$ is the midpoint of $$A B$$, $$E F \\bot A F$$, $$\\angle B A C = \\angle C A F$$, if $$B C = 2 \\sqrt{3}$$, $$A F = 1$$, then $$E F =$$          ."},{"type":"image_path","image_path":"images/8843_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"8844","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$\\text{Rt} \\triangle A B C$$, $$A B = B C$$, $$\\angle A B C = 90 \\circ$$, points $$D$$ and $$E$$ are moving points on sides $$A C$$ and $$B C$$ respectively, satisfying $$C E = \\sqrt{2} A D$$. Find the minimum value of $$\\frac{D E}{D B}$$."},{"type":"image_path","image_path":"images/8844_q0.png"}],"answer":"$$\\frac{\\sqrt{10} - \\sqrt{2}}{2}$$"}
{"id":"8852","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, points $$D$$ and $$E$$ are on sides $$BC$$ and $$AC$$ of $$\\triangle ABC$$, respectively. Connect $$DE$$. Fold $$\\triangle CDE$$ along $$DE$$ so that point $$C$$ lands at point $$F$$. Connect $$BF$$ and $$CF$$. If $$EF \\parallel AB$$, $$BF \\bot CF$$, $$AB = DF = 3\\sqrt{5}$$, $$FC = 12$$, $$DE = 5$$, then the length of $$AE$$ is      ."},{"type":"image_path","image_path":"images/8852_q0.png"}],"answer":"$$10 - 3 \\sqrt{5}$$"}
{"id":"8865","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\triangle ABC$$, $$AB = 4$$, $$\\angle ABC = 60^\\circ$$, $$\\angle ACB = 45^\\circ$$, D is the midpoint of $$BC$$, line $$l$$ passes through point D, from B draw $$BF \\bot l$$ to F, from A draw $$AE \\bot l$$ to E, find the maximum value of $$AE + BF$$ is      ."},{"type":"image_path","image_path":"images/8865_q0.png"}],"answer":"$$2 \\sqrt{6}$$"}
{"id":"8891","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that quadrilateral $$A B C D$$ is a square, $$A B = 3$$, and $$E$$ is a point on diagonal $$A C$$. Connect $$D E$$, and from point $$E$$ draw $$E F \\bot D E$$, intersecting the extension of $$B C$$ at point $$F$$. Construct rectangle $$D E F G$$ with $$D E$$ and $$E F$$ as adjacent sides. Connect $$C G$$, and given that $$\\tan \\angle C G F = \\frac{1}{3}$$, then $$A E =$$         ."},{"type":"image_path","image_path":"images/8891_q0.png"}],"answer":"$$2 \\sqrt{2}$$"}
{"id":"8897","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the square $$ABCD$$ with side length 4 is rotated counterclockwise about point $$A$$ to obtain the square $$AB^{'}C^{'}D^{'}$$. Connect $$BB^{'}$$ and $$BC^{'}$$. During the entire rotation process from 0° to 180°, when $$BB^{'} = BC^{'}$$, the area of $$\\triangle BB^{'}C^{'}$$ is            ."},{"type":"image_path","image_path":"images/8897_q0.png"}],"answer":"$$8 + 4 \\sqrt{3}$$ or $$8 - 4 \\sqrt{3}$$"}
{"id":"8911","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$BC$$ is the diameter of $$\\bigodot O$$ with radius 5, and A, D are two points on the circle. $$AE \\bot BC$$ at point E, $$DF \\bot BC$$ at point F, $$AE = 3$$, $$DF = 4$$. Point P is a moving point on $$BC$$. $$AP$$ intersects $$DE$$ at point Q. When $$PA + PD$$ is minimized, the length of $$AQ$$ is          ."},{"type":"image_path","image_path":"images/8911_q0.png"}],"answer":"$$\\frac{21}{11} \\sqrt{2}$$"}
{"id":"8921","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 3, AD = 4, point E is the midpoint of AD, point P is a moving point on BE, point Q is the midpoint of PC, connect AQ, then the minimum length of AQ is      ."},{"type":"image_path","image_path":"images/8921_q0.png"}],"answer":"$$\\frac{12 \\sqrt{13}}{13}$$"}
{"id":"8941","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point F is a point inside rectangle $$A B C D$$, and point E lies on side $$B C$$. Connect $$A F$$ and $$E F$$. Rotate segment $$A F$$ clockwise around point $$A$$ by $$90^\\circ$$ to obtain $$A P$$, and connect $$P E$$. If $$A B = 8$$, $$B C = 6$$, $$B E = \\frac{1}{2} C E$$, $$E F = 4$$, then the minimum value of segment $$P E$$ is          ."},{"type":"image_path","image_path":"images/8941_q0.png"}],"answer":"$$2 \\sqrt{34} - 4$$"}
{"id":"8944","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD, E is a moving point on CD. Fold the square along BE such that point C lands at point F. Connect CF and extend it to intersect AD at point G. Extend BF to intersect side AD at point H. If $$\\frac{C E}{B C}$$=$$\\frac{\\text{2}}{\\text{3}}$$, then the value of $$\\frac{G H}{D H}$$ is        ."},{"type":"image_path","image_path":"images/8944_q0.png"}],"answer":"$$\\frac{\\text{1}}{\\text{7}}$$"}
{"id":"8949","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$\\angle A = 30 \\circ$$, $$A C = 6$$, D is a moving point on AB, and an isosceles right triangle $$\\triangle D C E$$ is constructed with DC as the hypotenuse, such that $$\\angle C E D = 90 \\circ$$, and points E and A lie on opposite sides of CD. Connect BE; the minimum value of BE is      ."},{"type":"image_path","image_path":"images/8949_q0.png"}],"answer":"$$\\frac{\\sqrt{6}}{2}$$/$$\\frac{1}{2} \\sqrt{6}$$"}
{"id":"8982","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 60 \\circ$$, $$\\angle A B C < 60 \\circ$$, the three angle bisectors $$A D$$, $$B E$$, $$C F$$ intersect at point O, $$O H \\bot B C$$ at point H. The following conclusions: ① $$\\angle B O C = 120 \\circ$$, ② $$\\angle D O H = \\angle O C B - \\angle O B C$$, ③ $$O D$$ bisects $$\\angle B O C$$, ④ $$O E = O F$$, among which the correct conclusion numbers are               ."},{"type":"image_path","image_path":"images/8982_q0.png"}],"answer":"①②④"}
{"id":"8988","difficulty":"0.4","question_list":[{"type":"text","text":"Given that the side length of square $$A B C D$$ is 4, point $$P$$ is a moving point on segment $$A D$$ (not coinciding with point $$A$$), and point $$E$$ is the reflection of point $$A$$ across line $$B P$$. Connect $$P E$$, $$B E$$, $$C E$$, and $$D E$$. When $$\\triangle C D E$$ is an isosceles triangle with $$C E$$ as one of its equal sides, the value of $$A P$$ is      ."},{"type":"image_path","image_path":"images/8988_q0.png"}],"answer":"$$8 - 4 \\sqrt{3}$$ or $$\\frac{4 \\sqrt{3}}{3}$$"}
{"id":"8991","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, there is a square $$GICJ$$ at the bottom-right corner of the square $$ABCD$$. Construct a square $$EFGH$$ with vertex $$G$$ extending to the left, such that points $$E$$ and $$F$$ lie on sides $$AB$$ and $$BI$$, respectively. When points $$A$$, $$H$$, and $$J$$ are collinear, the value of $$\\frac{GF}{GI}$$ is       ."},{"type":"image_path","image_path":"images/8991_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"9004","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, construct a square $$A D E B$$ downward using the hypotenuse $$A B$$ as a side. Draw $$E F \\parallel B C$$ intersecting $$A C$$ at point F. Draw $$C G \\parallel B E$$ intersecting $$E F$$ at point G. Connect $$D G$$. If $$A F = 3$$ and $$D E = 15$$, then the quantitative relationship between segments $$A D$$ and $$C G$$ is                 ; the area of quadrilateral $$C G E B$$ is        ."},{"type":"image_path","image_path":"images/9004_q0.png"}],"answer":"$$A D \\parallel C G$$ and $$A D = C G$$     $$81$$"}
{"id":"9016","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 30 \\circ$$, $$A C = 4 \\sqrt{3}$$, $$A B = 8$$, point $$D$$ is inside $$\\triangle A B C$$, connect $$D A$$, $$D B$$, $$D C$$, then the minimum value of $$D C + D B + \\sqrt{3} A D$$ is      ."},{"type":"image_path","image_path":"images/9016_q0.png"}],"answer":"$$4 \\sqrt{13}$$"}
{"id":"9023","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2 B C$$, point $$M$$ is the midpoint of side $$C D$$, and points $$E, F$$ are points on sides $$A B$$ and $$B C$$ respectively, such that $$A F \\bot M E$$ at point $$G$$. If $$E B = 2$$ and $$B F = 1$$, then the area of quadrilateral $$B F G E$$ is      ."},{"type":"image_path","image_path":"images/9023_q0.png"}],"answer":"$$\\frac{85}{52}$$"}
{"id":"9027","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, AB = 6, BC = 4, ∠A = 120°, E is the midpoint of AB, and point F lies on the sides of parallelogram ABCD. If triangle AEF is an isosceles triangle, then the length of EF is     ."},{"type":"image_path","image_path":"images/9027_q0.png"}],"answer":"3$$\\sqrt{3}$$ or 3 or $$\\frac{\\sqrt{57}}{2}$$"}
{"id":"9038","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of a semicircle, $$AB = 10$$, the distance from point $$O$$ to chord $$AC$$ is $$4$$, point $$P$$ starts from $$B$$ and moves toward point $$A$$ along $$BA$$ at a speed of $$1$$ unit per second, connecting $$CP$$, after      seconds, $$\\Delta APC$$ is an isosceles triangle."},{"type":"image_path","image_path":"images/9038_q0.png"}],"answer":"$$\\frac{14}{5}$$ or $$4$$ or $$5$$"}
{"id":"9052","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta A B C$$, $$\\angle A C B = 45 \\circ$$, $$A B = 4$$, points $$E$$ and $$F$$ lie on sides $$B C$$ and $$A B$$ respectively, point $$E$$ is the midpoint of side $$B C$$, $$A B = 3 A F$$, connect $$A E$$ and $$C F$$ intersecting at point $$P$$, then the maximum area of $$\\Delta A B P$$ is   ."},{"type":"image_path","image_path":"images/9052_q0.png"}],"answer":"$$1 + \\sqrt{2}$$"}
{"id":"9055","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, OP bisects ∠AOB, ∠AOP = 15°, PC ∥ OA, PC = 4, and point D is a moving point on ray OA. Then the minimum value of PD is     ."},{"type":"image_path","image_path":"images/9055_q0.png"}],"answer":"2"}
{"id":"9078","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$ABCD$$ is a square, $$\\triangle DEF$$ rotates about point $$D$$ ($$DE < AB$$), $$\\angle EDF = 90^\\circ$$, $$DE = DF$$, connect $$AE$$, $$CF$$; line $$AE$$ intersects line $$CF$$ at point $$G$$. Connect $$BG$$. If $$AB = 6$$, $$DE = 2\\sqrt{2}$$, write down, during the rotation of $$\\triangle DEF$$,"},{"type":"image_path","image_path":"images/9078_q0.png"},{"type":"text","text":"① When point $$E$$ is inside square $$ABCD$$ and $$EF \\perp CD$$, $$BG =$$      ; ② The minimum length of segment $$BG$$      ."}],"answer":"$$\\frac{12 \\sqrt{10}}{5}$$/$$\\frac{12}{5} \\sqrt{10}$$     $$2 \\sqrt{14}$$"}
{"id":"9099","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ is a diagonal, $$A B = C D$$, $$\\angle D A C + \\angle B C A = 180 \\circ$$, $$\\angle B A C + \\angle A C D = 90 \\circ$$. If the area of quadrilateral $$A B C D$$ is 7, then the length of $$C D$$ is         ."},{"type":"image_path","image_path":"images/9099_q0.png"}],"answer":"$$\\sqrt{14}$$"}
{"id":"9111","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, segment AD is a median, point O is the midpoint of segment AD, line l passes through point O, and points B and C lie on the same side of l. Perpendiculars are drawn from points B, C, D, and A to line l, with feet of the perpendiculars being points E, F, H, and G, respectively. Which of the following statements are always correct?"},{"type":"image_path","image_path":"images/9111_q0.png"},{"type":"text","text":"① $$\\triangle A I G \\approx \\triangle B I E$$; ② $$A G = D H$$; ③ $$2 A G = B E + C F$$; ④ If points B and C lie on opposite sides of l, then $$2 A G = B E - C F$$."}],"answer":"②③/③②"}
{"id":"9113","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $$AC, BD$$ of square $$ABCD$$ intersect at point $$O$$, point $$F$$ is a point on $$CD$$, $$OE \\bot OF$$ intersects $$BC$$ at point $$E$$, connect $$AE, BF$$ intersecting at point $$P$$, connect $$OP$$. The following conclusions:"},{"type":"image_path","image_path":"images/9113_q0.png"},{"type":"text","text":"① $$AE \\bot BF$$; ② $$\\angle OPA = 45^\\circ$$; ③ $$AP - BP = \\sqrt{2} OP$$; ④ If $$BE : CE = 2 : 3$$, then $$\\tan \\angle CAE = \\frac{4}{7}$$; ⑤ The area of quadrilateral $$OECF$$ is $$\\frac{1}{4}$$ of the area of square $$ABCD$$. Among these, the correct conclusions are         ."}],"answer":"①②③⑤"}
{"id":"9121","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A C = 2$$, $$B C = 4$$, points E and F lie on $$A B$$ and $$A C$$ respectively, and the reflection of point A over line EF, denoted $$A^{'}$$, lies on $$B C$$. Let $$C A^{'} = x$$. If $$A E = A F$$, then x =       ; let $$A E = y$$, write the function expression of y in terms of x:           ."},{"type":"image_path","image_path":"images/9121_q0.png"}],"answer":"$$\\sqrt{5} - 1$$     $$y = \\frac{\\sqrt{5} x^{2} + 4 \\sqrt{5}}{4 x + 4}$$"}
{"id":"9132","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus $$A B C D$$, $$\\angle A = 60^\\circ$$, $$A D = 4$$, point M is a point on $$A D$$ such that $$A M = 3$$, and point P is a moving point on $$A B$$. Fold $$\\triangle A M P$$ along $$M P$$ to obtain $$\\triangle Q M P$$, where point A corresponds to point Q. Connect $$D Q$$. When point Q lies on a diagonal of rhombus $$A B C D$$, the length of $$D Q$$ is      ."},{"type":"image_path","image_path":"images/9132_q0.png"}],"answer":"$$\\frac{1 + \\sqrt{33}}{2}$$ or $$\\sqrt{7}$$"}
{"id":"9184","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure: In equilateral triangle ABC, $$AB = 1$$, E and F are moving points on sides $$AB$$ and $$AC$$ respectively, and $$CF = 2BE$$. Find the minimum value of $$BF + 2CE$$          ."},{"type":"image_path","image_path":"images/9184_q0.png"}],"answer":"$$\\sqrt{7}$$"}
{"id":"9191","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is 2, M is the midpoint of $$B C$$, and N is a moving point on $$A M$$. Through point N, draw $$E F \\bot A M$$, intersecting $$A B$$ and $$C D$$ at points E and F, respectively. If the value of $$E M + A F$$ is $$\\sqrt{10}$$, then the length of $$D F$$ is      ."},{"type":"image_path","image_path":"images/9191_q0.png"}],"answer":"$$\\frac{2}{3}$$"}
{"id":"9203","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point D is a point inside $$\\triangle A B C$$, $$B D$$ bisects $$\\angle A B C$$, $$C D \\bot B D$$ at point D, $$\\angle A C D = \\angle A$$. If $$c o s \\angle D C B = \\frac{1}{3}$$, $$A B = 5$$, then the length of $$B C$$ is          ."},{"type":"image_path","image_path":"images/9203_q0.png"}],"answer":"3"}
{"id":"9208","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square A B C D$$, $$A B = \\sqrt{2}$$, $$B C = \\sqrt{6}$$, $$\\angle A B C = 60^\\circ$$. Construct squares outwardly on each side of $$\\square A B C D$$, and connect the centers of these four squares in order to form quadrilateral $$E F G H$$. The area of quadrilateral $$E F G H$$ is      ."},{"type":"image_path","image_path":"images/9208_q0.png"}],"answer":"$$7$$"}
{"id":"9209","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus $$A B C D$$, $$\\angle B = 60 \\circ$$, E is a point on side $$A B$$, triangle $$\\triangle B C E$$ is folded along $$C E$$, and the image of point B is point F, $$C F$$ intersects $$A D$$ at G, $$E F$$ intersects $$A D$$ at H. If $$A B = 3 A E$$, then $$C G : F G =$$     ."},{"type":"image_path","image_path":"images/9209_q0.png"}],"answer":"$$7 : 1$$"}
{"id":"9213","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C$$, $$\\angle A B C = 45 \\circ$$, point $$D$$ is the midpoint of $$B C$$, $$C E \\bot A D$$ at point $$E$$, and its extension intersects $$A B$$ at point $$F$$, $$B M \\bot C F$$ intersects the extension of $$C F$$ at point $$M$$, $$C N$$ bisects $$\\angle A C B$$ and intersects $$A D$$ at point $$G$$, and intersects $$A B$$ at point $$N$$, connect $$D F$$, then the following conclusions: ① $$\\angle A D C = \\angle B D F$$; ② $$A D = C F + D F$$; ③ $$S_{\\triangle A C D} = 3 S_{\\triangle C D F}$$; ④ $$\\angle A G N = \\angle B F D$$; ⑤ $$A E - B M = E M$$; ⑥ $$\\angle A D F = 2 \\angle C A D$$. Among these, the correct ones are        . (Only fill in the serial numbers)"},{"type":"image_path","image_path":"images/9213_q0.png"}],"answer":"①②③④⑤⑥"}
{"id":"9218","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that the diagonals $$AC$$ and $$BD$$ of rectangle $$ABCD$$ intersect at point O, and point E is a moving point on side $$AB$$, with $$CE$$ intersecting $$BD$$ at point F, and connecting $$OE$$."},{"type":"image_path","image_path":"images/9218_q0.png"},{"type":"text","text":"(1) If point E is the midpoint of $$AB$$, then $$\\frac{OF}{FB}$$ =       ; (2) If point F is the midpoint of $$OB$$, then $$\\frac{AE}{BE}$$ =         ."}],"answer":"$$\\frac{1}{2}$$/0.5     2"}
{"id":"9244","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point E is inside $$\\square A B C D$$, EB⊥BC, ED⊥CD, and $$\\angle E A B = 45 \\circ$$. Connect CE. For the following four conclusions: ① $$\\angle A D E = \\angle A B E$$; ② $$\\angle D A E = \\angle D C E$$; ③ $$B E = A D$$; ④ When $$\\angle D A B = 60 \\circ$$, $$C E = 2 A E$$, the sequence numbers of all correct conclusions are        ."},{"type":"image_path","image_path":"images/9244_q0.png"}],"answer":"①②③④"}
{"id":"9246","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle $$\\triangle ABC$$, $$AB = BC = AC = 5$$, point $$M$$ is a point on the line containing the altitude $$AD$$ from side $$BC$$, and an equilateral triangle $$\\triangle BMN$$ is constructed with $$BM$$ as a side. Connect $$DN$$; then the minimum value of $$DN$$ is      ."},{"type":"image_path","image_path":"images/9246_q0.png"}],"answer":"$$\\frac{5}{4}$$"}
{"id":"9268","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 4 \\sqrt{2}$$, E is the midpoint of $$A B$$, and F is a moving point on side $$A D$$ (point F does not coincide with points A or D). Fold $$\\triangle A E F$$ along the line $$E F$$, and let the image of point A be $$A^{'}$$. Connect $$A^{'} D$$ and $$A^{'} C$$. When $$\\triangle A^{'} D C$$ is an isosceles triangle, the length of $$A F$$ is       ."},{"type":"image_path","image_path":"images/9268_q0.png"}],"answer":"$$\\sqrt{2}$$ or 2 or $$2 \\sqrt{2}$$"}
{"id":"9281","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, fold the paper triangle △ABC along MN such that point C lands inside quadrilateral ABNM; then there is a constant quantitative relationship among ∠1, ∠2, and ∠C. This relationship is   ."},{"type":"image_path","image_path":"images/9281_q0.png"}],"answer":"2∠C=∠1+∠2"}
{"id":"9292","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠BAC = 120°, AB = AC, point D is a point on side BC, point E is a point on segment CD, and CE = 1, AB = 2$$\\sqrt{3}$$, ∠DAE = 60°, then the length of DE is         ."},{"type":"image_path","image_path":"images/9292_q0.png"}],"answer":"$$\\frac{7}{3}$$"}
{"id":"9294","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, AB is the diameter of ⊙O, C is a point on the circle, and ∠AOC = 120°. The radius of ⊙O is 2. P is a moving point on the circle, and Q is the midpoint of AP. Then the maximum and minimum values of the length of CQ are     ."},{"type":"image_path","image_path":"images/9294_q0.png"}],"answer":"1+$$\\sqrt{7}$$"}
{"id":"9306","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 4, BC = $$4 \\sqrt{3}$$, point P moves along segment BC (including points B and C). Connect AP. Rotate segment AP counterclockwise 60° about point A to obtain AQ, and connect DQ. What is the minimum value of segment DQ?"},{"type":"image_path","image_path":"images/9306_q0.png"}],"answer":"2"}
{"id":"9323","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, AB is the diameter of ⊙O, and side DE of rectangle ACDE is tangent to ⊙O, with point C on ⊙O. If $$\\frac{A E}{D E} = \\frac{1}{4}$$, $$B D = \\frac{8}{3}$$, then $$A B =$$     ."},{"type":"image_path","image_path":"images/9323_q0.png"}],"answer":"$$\\frac{10}{3}$$"}
{"id":"9325","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, points E and F lie on sides $$A B$$ and $$B C$$ respectively, with $$A E = 2$$, $$F C = 5$$, and $$\\angle E D F = \\angle F D C$$. The side length of square $$A B C D$$ is      ."},{"type":"image_path","image_path":"images/9325_q0.png"}],"answer":"$$3 \\sqrt{5}$$"}
{"id":"9342","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A = 90 \\circ$$, $$A B = 6$$, $$A C = 3$$, $$D$$ is a point on side $$A B$$ such that $$A D = 2 B D$$, draw $$D E \\bot D C$$, intersecting $$B C$$ at point $$F$$, connect $$C E$$, if $$\\angle D C E = \\angle B$$, then the value of $$\\frac{E F}{D F}$$ is      ."},{"type":"image_path","image_path":"images/9342_q0.png"}],"answer":"$$\\frac{7}{4}$$"}
{"id":"9352","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given ∠MON = 30°, B is a point on OM, BA ⊥ ON at A, quadrilateral ABCD is a square, P is a moving point on ray BM, connect CP, rotate CP 90° clockwise around point C to obtain CE, connect BE. If AB = $$\\sqrt{3}$$, then the minimum value of BE is        ."},{"type":"image_path","image_path":"images/9352_q0.png"}],"answer":"$$\\frac{3 + \\sqrt{3}}{2}$$"}
{"id":"9362","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the right trapezoid $$A B C D$$, $$\\angle A = \\angle B = 90 \\circ$$, the angle bisectors of $$\\angle A D C$$ and $$\\angle B C D$$ intersect exactly at point $$E$$ on side $$A B$$. Rotate $$\\triangle C B E$$ counterclockwise about point $$E$$ to $$\\triangle E F G$$, such that point $$B$$ lands at point $$F$$ on segment $$E C$$, and point $$C$$ lands at point $$G$$. Let $$E G$$ and $$F G$$ intersect $$C D$$ at points $$M$$ and $$N$$, respectively. Given $$A B = 2$$ and $$B C = 2 \\sqrt{2}$$, find the value of $$D M : M N$$."},{"type":"image_path","image_path":"images/9362_q0.png"}],"answer":"$$1 : 2$$"}
{"id":"9363","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rectangle ABCD, point M is a point on the ray CD, AH ⊥ BM, with foot of perpendicular H. Fold △ABH along AB to obtain △ABH′. P and Q are the midpoints of AH′ and BH′, respectively. If AD = 2 and AB = 6, then the maximum value of PQ + CQ is              ."},{"type":"image_path","image_path":"images/9363_q0.png"}],"answer":"7"}
{"id":"9366","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠BCA = 90°, AB = $$\\sqrt{5}$$, AC = 2, D is a moving point on the hypotenuse AB (not coinciding with points A or B), DE ⊥ AC, DF ⊥ BC, with feet at E and F respectively. Connect EF; then the minimum value of EF is     ."},{"type":"image_path","image_path":"images/9366_q0.png"}],"answer":"$$\\frac{2 \\sqrt{5}}{5}$$"}
{"id":"9382","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, if point $$O$$ is a point outside the square $$A B C D$$, $$O D = 2 \\sqrt{2}$$, $$O C = 1$$, $$B O = \\sqrt{10}$$, then the tangent of $$\\angle C D O$$ is         ."},{"type":"image_path","image_path":"images/9382_q0.png"}],"answer":"$$\\frac{1}{3}$$"}
{"id":"9384","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given $$\\triangle A B C$$, $$\\triangle D C E$$, $$\\triangle F E G$$ are three congruent isosceles triangles, with bases $$B C$$, $$C E$$, $$E G$$ lying on the same straight line, and $$A B = \\sqrt{3}$$, $$B C = 1$$, $$A G$$ intersects $$D C$$, $$D E$$, $$E F$$ at points $$P$$, $$Q$$, $$R$$, respectively. Then $$P Q =$$      ."},{"type":"image_path","image_path":"images/9384_q0.png"}],"answer":"$$\\frac{1}{2}$$/0.5"}
{"id":"9386","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, two paper strips of equal width are crossed and overlapped. If the quadrilateral $$A B C D$$ formed by the overlapping region has $$A B = 3$$ and $$A C = 2$$, then the length of $$B D$$ is               ."},{"type":"image_path","image_path":"images/9386_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"9425","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rotate the equilateral triangle $$\\triangle A B C$$ clockwise around point C by 120° to obtain $$\\triangle E D C$$. Draw $$A N \\bot B C$$ intersecting $$B D$$ at F, and let point P be a point on $$C F$$. Connect $$A D$$, $$B D$$, $$C F$$, $$P A$$, and $$P B$$. The following conclusions hold:\n① The four sides of quadrilateral $$A C E D$$ are equal; ② $$B P$$ bisects $$\\angle D B C$$; ③ $$S_{\\triangle B D E} = 12 S_{\\triangle C N F}$$; ④ $$2 P B + P C > 6 F N$$. The correct conclusions are          ."},{"type":"image_path","image_path":"images/9425_q0.png"}],"answer":"①③/③①"}
{"id":"9432","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 30$$, point E lies on $$A D$$ such that $$D E = 6$$. Point G is the midpoint of $$A E$$, and point P is a moving point on side $$B C$$. F is the midpoint of $$E P$$. Find the minimum value of $$G F + E F$$        ."},{"type":"image_path","image_path":"images/9432_q0.png"}],"answer":"13"}
{"id":"9445","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A D = 2 A B$$, $$C E \\bot A B$$ at point E, points F and G are the midpoints of $$A D$$ and $$B C$$ respectively, and connect $$C F$$, $$E F$$, $$F G$$. The following four statements: ① $$C E \\bot F G$$; ② quadrilateral $$A B G F$$ is a rhombus; ③ $$B C = 2 E G$$; ④ $$\\angle D F C = \\angle E F G$$. The correct ones are         . (fill in the serial numbers)"},{"type":"image_path","image_path":"images/9445_q0.png"}],"answer":"①②③④"}
{"id":"9457","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90^\\circ$$, $$\\angle B = 30^\\circ$$, $$A C = 8$$. D is a moving point on $$B C$$, connect $$A D$$, the perpendicular bisector of $$A D$$ intersects $$A C$$ and $$A B$$ at points E and F, respectively. The maximum length of segment $$B F$$ is       ."},{"type":"image_path","image_path":"images/9457_q0.png"}],"answer":"$$\\frac{32}{3}$$"}
{"id":"9465","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point $$C$$ lies on segment $$AB$$, $$\\triangle DAC$$ is an equilateral triangle, and quadrilateral $$CDEF$$ is a square. Point $$P$$ is a moving point on segment $$AE$$. Connect $$PB$$ and $$PC$$. If $$AC = 2$$ and $$BC = 3$$, then the minimum value of $$PB + PC$$ is      ."},{"type":"image_path","image_path":"images/9465_q0.png"}],"answer":"$$\\sqrt{29}$$"}
{"id":"9472","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 8, B C = 12$$, points $$E$$, $$F$$, $$G$$ move along sides $$A D$$, $$B C$$, $$A B$$ respectively, and segment $$E F$$ always passes through the center of symmetry of the rectangle. The minimum perimeter of $$\\triangle E F G$$ is         ."},{"type":"image_path","image_path":"images/9472_q0.png"}],"answer":"$$4 \\sqrt{13} + 8$$"}
{"id":"9475","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\text{Rt} \\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, points $$D$$, $$E$$ lie on side $$B C$$ (point $$D$$ is to the left of point $$E$$), $$\\angle C A D = 2 \\angle B A D$$, $$A D = C E$$, point $$F$$ lies on side $$A C$$, $$\\angle B D A = \\angle E F A$$, if $$C F = 4$$, $$C D = a$$, $$E B = b$$, then $$A D =$$      . (Express in terms of $$a$$, $$b$$)"},{"type":"image_path","image_path":"images/9475_q0.png"}],"answer":"$$a + 2 - b$$"}
{"id":"9481","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the radius of ⊙O is 5, P is a point on the extension of diameter AB, BP = 1, CD is a chord of ⊙O with CD = 6. A parallelogram PCED is constructed with PC and PD as adjacent sides. When points C and D move on the circumference of the circle, the minimum length of segment PE is               ."},{"type":"image_path","image_path":"images/9481_q0.png"}],"answer":"4"}
{"id":"9484","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AD = 6, DC = 8, point H is on side AD, AH = 2, and E is a moving point on side AB. Connect HE. Construct a rhombus HEFG with HE as one side, located above and to the right of HE, such that point G lies on side DC. Connect CF."},{"type":"image_path","image_path":"images/9484_q0.png"},{"type":"text","text":"(1) When rhombus HEFG is a square, the length of DG is ; (2) As point E moves, the range of values for the area S of triangle FCG is ."}],"answer":"2     $$8 - 2 \\sqrt{13} \\leq S \\leq 8$$"}
{"id":"9497","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 1$$, $$B C = \\sqrt{3}$$, point $$M$$ is a moving point on diagonal $$A C$$, connect $$D M$$, then the minimum value of $$D M + \\frac{1}{2} A M$$ is      ."},{"type":"image_path","image_path":"images/9497_q0.png"}],"answer":"$$\\frac{3}{2}$$"}
{"id":"9502","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = B C$$, $$A D \\bot B C$$, $$C E \\bot A B$$, with feet at points $$D$$ and $$E$$ respectively. Point $$F$$ lies on the extension of $$D A$$, and connecting $$B F$$ intersects the extension of $$C E$$ at point $$M$$. $$A D = 2 C D$$.\n\n(1) If $$A E = 2$$, then $$B D =$$  ;\n\n(2) If $$B M : M F = 6 : 7$$, $$E M = 1$$, then $$A F =$$  ."},{"type":"image_path","image_path":"images/9502_q0.png"}],"answer":"3     $$\\frac{65}{18}$$/$$3 \\frac{11}{18}$$"}
{"id":"9517","difficulty":"0.4","question_list":[{"type":"text","text":"In $$\\text{Rt} \\Delta A B C$$, $$\\angle A = 90 \\circ$$, $$A B = \\sqrt{5}$$, $$A C = 2 \\sqrt{5}$$, $$M$$ is the midpoint of $$A C$$, and $$\\Delta A B C$$ is rotated counterclockwise about point $$M$$ to obtain $$\\Delta A^{'} B^{'} C^{'}$$. During the rotation, the lines $$B B^{'}$$ and $$C C^{'}$$ intersect at point $$P$$. Then the maximum value of $$B P$$ is           ."},{"type":"image_path","image_path":"images/9517_q0.png"}],"answer":"$$5 \\sqrt{2}$$"}
{"id":"9534","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = a$$, $$B E = 6$$, $$E C = 10$$, fold $$\\triangle A B E$$ along $$A E$$; if the image point F of point B falls exactly on the diagonal of rectangle $$A B C D$$, then the value of $$a$$ is               ."},{"type":"image_path","image_path":"images/9534_q0.png"}],"answer":"12 or $$4 \\sqrt{6}$$"}
{"id":"9536","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$AE$$ is the angle bisector of $$\\angle CAM$$, point $$B$$ lies on ray $$AM$$, $$DE$$ is the perpendicular bisector of segment $$BC$$ intersecting $$AE$$ at $$E$$, and from point $$E$$, a perpendicular is drawn to $$AM$$ intersecting $$AM$$ at point $$F$$. If $$\\angle ACB = 28^\\circ$$ and $$\\angle EBD = 25^\\circ$$, then $$\\angle AED =$$ $$^\\circ$$."},{"type":"image_path","image_path":"images/9536_q0.png"}],"answer":"$$37$$"}
{"id":"9581","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus ABCD with side length 1, $$\\angle ABC = 60^\\circ$$. Translate $$\\triangle ABD$$ along the direction of ray BD to obtain $$\\triangle A'B'D'$$. Connect $$A'C$$, $$A'D$$, and $$B'C$$. The minimum value of $$A'C + B'C$$ is      ."},{"type":"image_path","image_path":"images/9581_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"9583","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 4$$. Fold $$\\triangle B C D$$ along diagonal $$B D$$ so that point C lands at $$C^{'}$$. Let $$B C^{'}$$ intersect $$A D$$ at point G. Points E and F lie on $$C^{'} D$$ and $$B D$$ respectively. Segment $$E F$$ intersects $$A D$$ at points H and M. Fold $$\\triangle F D E$$ along $$E F$$ so that point D lands at point A. Let segment $$A E$$ intersect $$B C^{'}$$ at point M. Find the length of segment $$C^{'} M$$."},{"type":"image_path","image_path":"images/9583_q0.png"}],"answer":"$$\\frac{308}{527}$$"}
{"id":"9585","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given $$\\triangle A B C$$, $$A B = A C$$, $$B C = 6$$, $$\\angle B A C = 120^\\circ$$, point D lies on $$B C$$ (not coinciding with B or C), connect $$A D$$, and reflect $$\\triangle A B D$$ and $$\\triangle A C D$$ over lines $$A B$$ and $$A C$$ to obtain $$\\triangle A B F$$ and $$\\triangle A C E$$, respectively. Connect $$E F$$. The following conclusions are given: ① $$E F = \\sqrt{3} A F$$; ② When $$A D \\bot A F$$, the length of $$C D$$ is $$2 \\sqrt{3}$$; ③ When points D, A, and F are collinear, quadrilateral $$A D C E$$ is a rhombus; ④ The minimum area of $$\\triangle A E F$$ is $$\\frac{3 \\sqrt{3}}{4}$$. The correct conclusions are ______. (Fill in the serial numbers)"},{"type":"image_path","image_path":"images/9585_q0.png"}],"answer":"①②③④"}
{"id":"9590","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus $$A B C D$$, $$A B = 9$$, $$\\angle A B C = 60 \\circ$$, point $$E$$ lies on side $$A B$$ such that $$B E = 2 A E$$, and point $$P$$ moves along side $$B C$$. Connect $$P E$$, and rotate segment $$P E$$ clockwise by $$60 \\circ$$ about point $$P$$ to obtain segment $$P F$$. Connect $$A F$$. What is the minimum length of segment $$A F$$?"},{"type":"image_path","image_path":"images/9590_q0.png"}],"answer":"3$$\\sqrt{3}$$"}
{"id":"9595","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, AB = 12, BC = 16, AC and BD intersect at O, E is a point on DC, and from point O, draw OF ⊥ OE intersecting BC at F. Let d = $$\\sqrt{DE^{2} + BF^{2}}$$, then the minimum value of d is      ."},{"type":"image_path","image_path":"images/9595_q0.png"}],"answer":"10"}
{"id":"9610","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\triangle A B C$$ is an isosceles right triangle, $$\\angle A B C = 90 ^{\\circ}$$, points $$D$$, $$E$$ move along sides $$A C$$, $$B C$$ respectively, connecting $$A E$$, $$B D$$ intersecting at point $$F$$, and always satisfying $$C E = \\sqrt{2} A D$$. Which of the following conclusions are correct: ① $$\\frac{B D}{A E} = \\frac{\\sqrt{2}}{2}$$; ② $$\\angle D F E = 120^{\\circ}$$; ③ $$\\sqrt{2} B D \\cdot B F = A C \\cdot B E$$. The correct ones are                     ."},{"type":"image_path","image_path":"images/9610_q0.png"}],"answer":"①"}
{"id":"9617","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, BD is the diagonal. Fold rectangle ABCD along the lines BE and BF such that point A lands at point M on BD, and point C lands at point N on BD. Connect EF. Given that AB = 6 and BC = 8, the length of EF is                  ."},{"type":"image_path","image_path":"images/9617_q0.png"}],"answer":"$$\\frac{5 \\sqrt{13}}{3}$$"}
{"id":"9632","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the radius of sector $$A O B$$ is 8, $$\\angle A O B = 90 \\circ$$, points C, D, E are moving points on arc $$A B$$, radius $$O A$$, and $$O B$$ respectively, $$\\angle C E D = 90 \\circ$$, $$\\text{tan} \\angle C D E = \\frac{2}{3}$$, then the minimum perimeter of $$\\triangle C D E$$ is        ."},{"type":"image_path","image_path":"images/9632_q0.png"}],"answer":"$$\\frac{40 \\sqrt{13}}{13} + 8$$"}
{"id":"9633","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 2 \\sqrt{3}$$, point $$E$$ lies on side $$A D$$, $$E D = 2$$, an arc is drawn with center at point $$E$$ and radius equal to $$A E$$, intersecting $$B C$$ at point $$F$$, and passing exactly through point $$C$$. Connect $$A C$$ and $$C E$$. The area of the shaded region is      ."},{"type":"image_path","image_path":"images/9633_q0.png"}],"answer":"$$\\frac{16 \\pi}{3} - 4 \\sqrt{3}$$"}
{"id":"9637","difficulty":"0.4","question_list":[{"type":"text","text":"Given ∠AOB = 45°, point P is inside ∠AOB, point P1 is symmetric to point P with respect to OA, point P2 is symmetric to point P with respect to OB, connect P1P2 intersecting OA and OB at E and F respectively. If P1E = $$\\frac{1}{2}$$ and OP = $$\\sqrt{2}$$, then the length of EF is     ."},{"type":"image_path","image_path":"images/9637_q0.png"}],"answer":"$$\\frac{5}{6}$$"}
{"id":"9691","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 2$$, point $$E$$ is a point on diagonal $$AC$$ such that $$AE = CB$$. Connect $$DE$$ and extend it to intersect $$BC$$ at point $$G$$. From point $$A$$, draw $$AH \\bot BE$$ meeting at point $$H$$, and intersect $$BC$$ at point $$F$$. Among the following conclusions: ① $$\\triangle ABF \\sim \\triangle DCG$$; ② $$\\angle BEG = 45^\\circ$$; ③ $$4BH^2 = BG \\cdot CD$$; ④ $$BF = 2\\sqrt{2} - 1$$. The correct conclusions are         ."},{"type":"image_path","image_path":"images/9691_q0.png"}],"answer":"①②③"}
{"id":"9731","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the radius of semicircle O is 2, and E is a point on the semicircle. Fold point E onto diameter AB (with EE′ ⊥ AB). When the folded arc intersects diameter AB at least at one point, the range of possible lengths for the crease CD is                 ."},{"type":"image_path","image_path":"images/9731_q0.png"}],"answer":"$$2 \\sqrt{3} \\leq C D \\leq 4$$"}
{"id":"9738","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 120 \\circ$$, $$A C = 8$$, $$B C = 4$$. Fold side $$B C$$ along $$C E$$ such that point B lands at point D on $$A B$$, then fold side $$A C$$ along $$C F$$ such that point A lands at point $$A^{'}$$ on the extension of $$C D$$. The two creases intersect the hypotenuse $$A B$$ at points E and F, respectively. The length of segment $$F A^{'}$$ is        ."},{"type":"image_path","image_path":"images/9738_q0.png"}],"answer":"$$\\frac{8 \\sqrt{7}}{7}$$"}
{"id":"9760","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, BP bisects ∠ABC, $$\\text{PD} \\bot \\text{BC}$$, and E, F are points on the two sides of the angle. There are four conclusions; those that can definitely lead to the remaining conclusions are: ① $$\\angle \\text{ABC} + \\angle \\text{EPF} = 180 \\circ$$; ② $$\\angle \\text{BEP} = \\angle \\text{PFC}$$; ③ $$\\text{PE} = \\text{PF}$$; ④ $$2 \\text{BD} = \\text{BF} + \\text{BE}$$,     ."},{"type":"image_path","image_path":"images/9760_q0.png"}],"answer":"From (1), (2), and (4), the other three can be derived."}
{"id":"9766","difficulty":"0.4","question_list":[{"type":"text","text":"△ABC is an equilateral triangle with side length 2, and point P is a moving point on line BC. Rotate segment AP counterclockwise 60° about point A to obtain AE. O is a moving point on side AB. Then the minimum value of OE is    ."},{"type":"image_path","image_path":"images/9766_q0.png"}],"answer":"$$\\sqrt{3}$$."}
{"id":"9767","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, point $$E$$ lies on diagonal $$BD$$ such that $$BE = BC$$. Connect $$CE$$ and extend it to intersect $$AD$$ at point $$F$$. Connect $$AE$$, and from point $$B$$, draw $$BG \\bot AE$$ intersecting at point $$G$$. Extend $$BG$$ to intersect $$AD$$ at point $$H$$. Among the following conclusions: ① $$AH = DF$$; ② $$\\angle AEF = 45^\\circ$$; ③ $$AH = DE$$; ④ $$S_{\\text{quadrilateral } EFHG} = S_{\\triangle DEF} + S_{\\triangle AGH}$$, the correct conclusions are      $$\\left(\\right.$$fill in the correct serial numbers$$\\left.\\right)$$"},{"type":"image_path","image_path":"images/9767_q0.png"}],"answer":"①②③"}
{"id":"9768","difficulty":"0.4","question_list":[{"type":"text","text":"In the rhombus $$A B C D$$, $$\\angle A = 120^\\circ$$, $$A B = 4$$, there are two moving points E and F on sides $$A D$$ and $$C D$$ respectively, always satisfying $$D E = D F$$. Connect $$B E$$ and $$E F$$, take the midpoint G of $$B E$$ and connect $$F G$$. Then the minimum value of $$F G$$ is     ."},{"type":"image_path","image_path":"images/9768_q0.png"}],"answer":"3"}
{"id":"9774","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point O is a point inside triangle $$A B C$$, $$O A = O B = O C = 4, \\angle B A C = 45 \\circ$$, and it is given that $$S_{\\triangle A O C} - S_{\\triangle A O B} = 2$$, then $$\\angle B O C =$$        , $$S_{\\triangle A B C} =$$           ."},{"type":"image_path","image_path":"images/9774_q0.png"}],"answer":"$$90 \\circ$$     $$2 \\sqrt{31} + 8$$"}
{"id":"9781","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\angle B A C =\\angle D A F =\\text{90}°$$, $$A B = A C$$, $$A D = A F$$, $$A B$$ and $$F E$$ intersect at point $$M$$, points $$D$$, $$E$$ are two points on side $$B C$$, and $$\\angle D A E =\\text{45}°$$, connect $$E F$$, $$B F$$, then the following conclusions: ① $$\\triangle A F B ≌\\triangle A D C$$; ② $$B E^{\\text{2}} + D C^{\\text{2}} = D E^{\\text{2}}$$; ③ $$A B - A D = E D - B E$$; ④ only when $$\\angle A M E =\\text{90}°$$, $$B F = B E$$, among which the correct ones are    . (fill in the serial numbers)"},{"type":"image_path","image_path":"images/9781_q0.png"}],"answer":"①②④"}
{"id":"9782","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A B = 10 \\text{cm}$$, $$A C = 8 \\text{cm}$$, $$O$$ is the midpoint of $$A B$$, point $$P$$ starts from point $$C$$ and moves toward point $$B$$ along side $$C B$$ at a speed of $$1 \\text{cm}/\\text{s}$$, and point $$Q$$ starts from point $$A$$ and moves toward point $$C$$ along side $$A C$$ at a speed of $$2 \\text{cm}/\\text{s}$$. Points $$P$$ and $$Q$$ start simultaneously, and when one point reaches its endpoint, the other point also stops moving simultaneously.\n(1) After 2 seconds of departure, the distance between points $$P$$ and $$Q$$ is                  $$\\text{cm}$$.\n(2) When the area of $$\\triangle O P Q$$ is minimized, the distance between points $$P$$ and $$Q$$ is                     $$\\text{cm}$$."},{"type":"image_path","image_path":"images/9782_q0.png"}],"answer":"$$2 \\sqrt{5}$$     $$\\frac{\\sqrt{61}}{2}$$"}
{"id":"9786","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, AB = AD = 2, ∠DAB = ∠DCA = 60°, then the maximum value of $$CD + CB$$ is          ."},{"type":"image_path","image_path":"images/9786_q0.png"}],"answer":"$$\\frac{4 \\sqrt{3}}{3}$$"}
{"id":"9788","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral ABCD, ∠A = 30°, AB = 8, point O is the midpoint of AB, OC = OD = $$\\frac{1}{2}$$AB, OC ∥ AD, point P is a moving point on AB (not coinciding with points A or B), then the minimum value of PD + PC is      ."},{"type":"image_path","image_path":"images/9788_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"9789","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, D is the midpoint of side BC. Connect AD, and fold △ACD along AD to obtain △ADF. DF intersects AB at point E. Connect BF, with BD = BF = 2 and AD = 3."},{"type":"image_path","image_path":"images/9789_q0.png"},{"type":"text","text":"(1) Connect CF; the measure of ∠DCF is ______ degrees; (2) The distance from point D to AF is ______."}],"answer":"30     $$\\frac{3 \\sqrt{21}}{7}$$"}
{"id":"9793","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = 5$$, $$B C = 4 \\sqrt{3}$$, $$\\angle A C B = 60 \\circ$$, D is a moving point inside $$\\triangle A B C$$, $$\\bigodot O$$ is the circumcircle of $$\\triangle A C D$$, intersecting $$B C$$ at point E, line $$B D$$ intersects $$\\bigodot O$$ at point P, $$\\overset{⌢}{A E} = \\overset{⌢}{C P}$$, then the minimum value of $$A D$$ is      ."},{"type":"image_path","image_path":"images/9793_q0.png"}],"answer":"$$\\sqrt{41} - 4$$"}
{"id":"9794","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a rectangle, point $$F$$ is a trisection point of side $$A B$$, with $$B F = 2 A F$$, point $$E_{1}$$ is the midpoint of side $$C B$$, connecting $$E_{1} F$$ and $$E_{1} D$$ to form $$\\triangle E_{1} F D$$; point $$E_{2}$$ is the midpoint of $$C E_{1}$$, connecting $$E_{2} F$$ and $$E_{2} D$$ to form $$\\triangle E_{2} F D$$; point $$E_{3}$$ is the midpoint of $$C E_{2}$$, connecting $$E_{3} F$$ and $$E_{3} D$$ to form $$\\triangle E_{3} F D$$; … continuing in this pattern, if the area of rectangle $$A B C D$$ is 6, then the area of $$\\triangle E_{2022} F D$$ is       ."},{"type":"image_path","image_path":"images/9794_q0.png"}],"answer":"$$3 - \\frac{1}{2^{2022}}$$"}
{"id":"9797","difficulty":"0.4","question_list":[{"type":"text","text":"The ancient Chinese mathematician Zhao Shuang, while annotating the \"Zhou Bi Suan Jing,\" used four congruent right triangles to form a square (as shown in the figure) and employed it to prove the Pythagorean theorem; this diagram is known as the \"Zhao Shuang Chord Diagram.\" Draw $$EM \\parallel NG \\parallel AD$$, and if $$GF = 2FM$$, then the value of $$\\frac{MN}{FD}$$ is        ."},{"type":"image_path","image_path":"images/9797_q0.png"}],"answer":"$$\\frac{\\sqrt{5}}{2}$$/$$\\frac{1}{2} \\sqrt{5}$$"}
{"id":"9803","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E is a moving point on $$CD$$ (point E does not coincide with points C or D). Connect $$AE$$ intersecting $$BD$$ at F. From F, draw $$FH \\bot AE$$ intersecting $$BC$$ at point H. From H, draw $$HG \\bot BD$$ at G. The following conclusions are given: ① $$AF = FH$$; ② $$\\angle HAE = 45^\\circ$$; ③ $$BD = 2FG$$; ④ The perimeter of $$\\triangle CEH$$ is 9. Among these, the ones that must be true are   ."},{"type":"image_path","image_path":"images/9803_q0.png"}],"answer":"①②③"}
{"id":"9804","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the isosceles right triangle $$\\triangle A B C$$, $$\\angle B A C = 90 \\circ$$, $$A B = A C$$, the altitude $$A H$$ intersects the median $$B D$$ at point F. From A, draw $$A E \\bot B D$$ intersecting BC at point E, and connect $$H D$$, yielding the following five conclusions: ① $$\\angle A B D = \\angle C A E$$, ② $$\\triangle A B F \\approx \\triangle C A E$$, ③ $$\\angle E D C - \\angle C B D = 30 \\circ$$, ④ $$A E + D E < A H + D H$$, ⑤ $$S_{\\triangle A B C} = 6 S_{\\triangle C D E}$$. Among these, the correct conclusions are      (fill in the serial numbers)."},{"type":"image_path","image_path":"images/9804_q0.png"}],"answer":"①②④⑤"}
{"id":"9808","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 1$$, $$B C = \\sqrt{3}$$, $$E$$ is a moving point on side $$A D$$, connect $$B E$$, $$A F \\bot B E$$ at $$F$$, $$G$$ is the midpoint of $$B C$$, connect $$F G$$, construct an equilateral triangle $$\\triangle F G H$$ to the right using $$F G$$ as a side, connect $$C H$$, then the minimum value of $$C H$$ is     ."},{"type":"image_path","image_path":"images/9808_q0.png"}],"answer":"$$\\frac{\\sqrt{7} - 1}{2}$$"}
{"id":"9823","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square ABCD is 1, and AC and BD are its diagonals. Rotate triangle DCB clockwise around point D by 45° to obtain triangle DGH. HG intersects AB at point E, connect DE intersecting AC at point F, and connect FG. The following conclusions are given: ① Quadrilateral AEGF is a rhombus; ② The area of triangle HED is $1 - \\frac{\\sqrt{2}}{2}$; ③ ∠AFG = 135°; ④ BC + FG = $\\sqrt{3}$. Among these, the correct conclusions are ______. (Fill in the correct sequence numbers)"},{"type":"image_path","image_path":"images/9823_q0.png"}],"answer":"①②③"}
{"id":"9829","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C \\bot A B, B D \\bot C D, B D = C D$$, if $$A B = m, A C = n$$, and $$m < n$$, then the length of $$A D$$ is          (expressed in terms of $$m, n$$)."},{"type":"image_path","image_path":"images/9829_q0.png"}],"answer":"$$\\frac{\\sqrt{2} \\left(n - m\\right)}{2}$$"}
{"id":"9888","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A D = 8 \\text{cm}$$, $$A B = 4 \\text{cm}$$, $$A E$$ bisects $$\\angle B A D$$ and intersects side $$B C$$ at point $$E$$, and intersects the extension of $$D C$$ at point $$F$$. The following conclusions: ① $$C E = 4 \\text{cm}$$; ② segments $$A F$$ and $$B C$$ bisect each other; ③ $$A C \\bot D F$$; ④ $$D E \\bot A F$$; among these, the correct conclusions are:      (fill in the serial numbers)."},{"type":"image_path","image_path":"images/9888_q0.png"}],"answer":"①②④"}
{"id":"9901","difficulty":"0.4","question_list":[{"type":"text","text":"Given: As shown in the figure, ∠ABC = 40°, point P is a moving point on ray BC. Triangle ABP is folded along AP, and the image of point B is point D. When line AD is perpendicular to BC, ∠ABD =     °."},{"type":"image_path","image_path":"images/9901_q0.png"}],"answer":"65 or 25."}
{"id":"9953","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, points D and E are points on sides AB and AC respectively, and BE = CD, CD ⊥ BE. If ∠A = 30°, BD = 1, CE = 2$$\\sqrt{3}$$, then the area of quadrilateral CEDB is     ."},{"type":"image_path","image_path":"images/9953_q0.png"}],"answer":"$$\\frac{19}{4}$$"}
{"id":"9967","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the right triangle $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, a square $$A B D E$$ is constructed outwardly on the hypotenuse $$A B$$, and the diagonals of the square intersect at point $$O$$. Connect $$O C$$. Given $$A C = 5$$ and $$O C = 6 \\sqrt{2}$$, the length of the other leg $$B C$$ is           ."},{"type":"image_path","image_path":"images/9967_q0.png"}],"answer":"$$7$$"}
{"id":"9972","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$. The vertices $$E$$, $$F$$, $$G$$, $$H$$ of rhombus $$E F G H$$ lie on sides $$A D$$, $$A B$$, $$B C$$, $$C D$$ respectively. If $$A F : B G = 3 : 4$$, then the length of $$C G$$ is        ."},{"type":"image_path","image_path":"images/9972_q0.png"}],"answer":"$$\\frac{8}{3}$$"}
{"id":"9980","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$ABCD$$, $$\\angle A = 60^\\circ$$, points M and N are any two points on sides $$AD$$ and $$AB$$, respectively. The rhombus $$ABCD$$ is folded along $$MN$$, and point A lands exactly at point E on diagonal $$BD$$. The following conclusions: ① $$\\triangle MED \\sim \\triangle ENB$$; ② If $$\\angle DME = 20^\\circ$$, then $$\\angle ENB = 100^\\circ$$; ③ If $$DE : BE = 1 : 2$$, then $$AM : AN = 1 : 2$$; ④ If the side length of the rhombus is 4, M is the midpoint of $$AD$$, and segment $$MC$$ is connected, then $$MC = 2\\sqrt{7}$$. The correct conclusions are:       (fill in all correct conclusion numbers)"},{"type":"image_path","image_path":"images/9980_q0.png"}],"answer":"①②④"}
{"id":"9988","difficulty":"0.4","question_list":[{"type":"text","text":"In isosceles triangle △ABC, AB = AC = 4, ∠BAC = 120°. A line l rotates arbitrarily about vertex A. From point C, drop a perpendicular to line l, with foot at H. The range of the distance between points B and H is     ."},{"type":"image_path","image_path":"images/9988_q0.png"}],"answer":"$$2 \\sqrt{7} - 2$$≤BH≤$$2 \\sqrt{7} + 2$$."}
{"id":"10001","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rectangle $$A B C D$$ is inscribed in $$\\bigodot O$$, $$A B = 40$$, $$B C = 55$$, $$E$$ is a point on arc $$B C$$, connect $$A E$$, $$C E$$, $$C E = 20$$, then $$A E =$$      ; $$F$$ is a moving point on ray $$C E$$, $$\\angle B F E > 90 \\circ$$, when $$B F = A B$$, $$E F =$$      ."},{"type":"image_path","image_path":"images/10001_q0.png"}],"answer":"$$65$$     $$24 - \\sqrt{511}$$/$$- \\sqrt{511} + 24$$"}
{"id":"10007","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$A C \\bot B C$$, $$D C \\bot E C$$, $$A C = B C$$, $$D C = E C$$, if $$A C = 2$$, $$C E = 3$$, then $$A D^{2} + B E^{2} =$$      ."},{"type":"image_path","image_path":"images/10007_q0.png"}],"answer":"26"}
{"id":"10020","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, ∠BAD = 45°, AB = 10. The paper is folded such that the image point A' of point A falls on side BC; the crease EF intersects AB, AD, and AA' at points E, F, and G, respectively. The paper is then folded again such that the image point C' of point C falls on A'F. Connect GC'. The distance from point G to AD is  , and the minimum value of GC' is     ."},{"type":"image_path","image_path":"images/10020_q0.png"}],"answer":"$$\\frac{5 \\sqrt{2}}{2}$$     $$\\frac{5 \\sqrt{2}}{2}$$"}
{"id":"10030","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of square $$A B C D$$ is $$3$$, points $$P$$ and $$Q$$ lie on the extensions of $$A B$$ and $$B C$$ respectively, and $$B P = C Q$$. Connect $$A Q$$ and $$D P$$, intersecting at point $$O$$, and intersecting sides $$C D$$ and $$B C$$ at points $$F$$ and $$E$$ respectively. Connect $$A E$$. The following conclusions are given: ① $$A Q \\bot D P$$; ② $$O A^{2} = O D \\cdot O P$$; ③ $$S_{\\triangle A O D} = S_{O E C F}$$; ④ When $$B P = 1$$, $$t a n \\angle O A E = \\frac{11}{16}$$. Among these, the correct ones are      . (Write all the correct conclusion numbers)"},{"type":"image_path","image_path":"images/10030_q0.png"}],"answer":"①②③"}
{"id":"10032","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, on the sides $$AC$$ and $$BC$$ of the equilateral triangle $$ABC$$, take points $$P$$ and $$Q$$ such that $$AP = CQ$$. Let $$AQ$$ and $$BP$$ intersect at point $$O$$. If $$BO = 6$$ and $$PO = 2$$, then the length of $$AP$$ is    , and the length of $$AO$$ is       ."},{"type":"image_path","image_path":"images/10032_q0.png"}],"answer":"4     $$1 + \\sqrt{13}$$"}
{"id":"10048","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square ABCD with side length 1, rotate ray AC around point A clockwise by α degrees (0 < α ≤ 360°) to obtain ray AE. Let M be the symmetric point of D with respect to ray AE. Then the minimum value of the length of segment CM is          ."},{"type":"image_path","image_path":"images/10048_q0.png"}],"answer":"$$\\sqrt{2}$$﹣1"}
{"id":"10056","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, rotate $$\\triangle A D C$$ counterclockwise about point D by $$90 \\circ$$ to obtain $$\\triangle F D E$$, such that points B, F, and E lie on the same straight line. Let $$A C$$ intersect $$B E$$ at point G, and connect $$D G$$. Which of the following conclusions is correct:     .\n①$$A C \\bot B E$$; ②$$\\triangle B C G \\sim \\triangle G A D$$; ③$$D F 2 = F C \\cdot D C$$; ④$$C G + \\sqrt{2} D G = E G$$."},{"type":"image_path","image_path":"images/10056_q0.png"}],"answer":"①③④"}
{"id":"10058","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the square $$ABCD$$ with side length 6, point $$E$$ is on diagonal $$AC$$. Connect $$DE$$, and from point $$E$$ draw a perpendicular to $$DE$$ intersecting the line containing side $$AB$$ at point $$F$$. Connect $$DF$$, intersecting the line containing diagonal $$AC$$ at point $$G$$. If $$AF = \\frac{1}{2} AB$$, then the length of segment $$EC =$$     ."},{"type":"image_path","image_path":"images/10058_q0.png"}],"answer":"$$\\frac{3}{2} \\sqrt{2}$$ or $$\\frac{9 \\sqrt{2}}{2}$$"}
{"id":"10118","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, through a point $P$ on the side $AB$ of the equilateral triangle $\\triangle ABC$ with side length 4, draw $PE \\bot AC$ at $E$. Let $Q$ be a point on the extension of $BC$ such that $PA = CQ$. Connect $PQ$ intersecting side $AC$ at $D$. Then the length of $DE$ is           ."},{"type":"image_path","image_path":"images/10118_q0.png"}],"answer":"$$2$$"}
{"id":"10144","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, parallelogram paper $$ABCD$$ is folded along crease $$MN$$, with point $$M$$ and point $$N$$ on sides $$AD$$ and $$BC$$ respectively. The images of points $$C$$ and $$D$$ are $$E$$ and $$F$$, respectively, and point $$F$$ lies inside the parallelogram. The extension of $$MF$$ intersects $$BC$$ at point $$G$$. $$EF$$ intersects side $$BC$$ at point $$H$$. Given $$AB = 6$$, $$\\angle B = 45^\\circ$$, $$CN = 2\\sqrt{2}$$, and when point $$H$$ is a trisection point of $$NG$$, the length of $$MD$$ is          ."},{"type":"image_path","image_path":"images/10144_q0.png"}],"answer":"$$6 \\sqrt{5} - 4 \\sqrt{2}$$ or $$3 \\sqrt{10} - \\sqrt{2}$$"}
{"id":"10149","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point E is a point on side BC of square ABCD. Triangle ABE is folded over AE as the axis of symmetry to obtain triangle AFE. Then, AD is folded onto AF, and the crease intersects the extension of BF at point H. BH intersects AE at point G. Connect DH.\n(1) The measure of ∠AHB is      ;\n(2) If AB = 2, the maximum distance from point H to AB is      ."},{"type":"image_path","image_path":"images/10149_q0.png"}],"answer":"45°     $$1 + \\sqrt{2}$$/$$\\sqrt{2} + 1$$"}
{"id":"10150","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the side length of rhombus ABCD is 4, ∠BAD = 60°, point E is a moving point on AD (not coinciding with A or D), point F is a moving point on CD, and AE + CF = 4. The minimum area of △BEF is     ."},{"type":"image_path","image_path":"images/10150_q0.png"}],"answer":"3$$\\sqrt{3}$$."}
{"id":"10159","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point O is a point inside the equilateral triangle $$\\triangle ABC$$, with $$OA = 3$$, $$OB = 4$$, $$OC = 5$$. Segment $$BO$$ is rotated counterclockwise by $$60^\\circ$$ about point B to obtain segment $$BO^{'}$$. The following conclusions: ① $$\\triangle BO^{'}A$$ can be obtained by rotating $$\\triangle BOC$$ counterclockwise about point B by $$60^\\circ$$; ② the distance between point O and $$O^{'}$$ is 4; ③ $$\\angle AOB = 150^\\circ$$; ④ $$S_{\\text{quadrilateral } AOBO^{'}} = 6 + 3\\sqrt{3}$$. Among these, the correct conclusions are            ."},{"type":"image_path","image_path":"images/10159_q0.png"}],"answer":"①②③"}
{"id":"10161","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 7$$, $$B C = \\sqrt{15}$$, point D lies on side $$A B$$, point E lies on side $$A C$$, $$C E = 1$$, fold quadrilateral $$B C E D$$ along line $$D E$$ to obtain quadrilateral $$G F E D$$, with point B mapped to point G, connect $$A G$$ and $$C F$$, when $$A G$$ is minimized, the value of $$\\cos \\angle E F C$$ is           ."},{"type":"image_path","image_path":"images/10161_q0.png"}],"answer":"$$\\frac{\\sqrt{10}}{4}$$"}
{"id":"10163","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, ∠ACB = 90°, AB = 4, point O is the midpoint of AB. An isosceles right triangle BCD is constructed outwardly with BC as one leg. Connect OD. When OD reaches its maximum value, the measure of ∠ODB is     ."},{"type":"image_path","image_path":"images/10163_q0.png"}],"answer":"22.5°"}
{"id":"10171","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, segment AB = 4, M is the midpoint of AB, and point P moves such that its distance to point M is 1. Connect PB; rotate segment PB 90° counterclockwise about point P to obtain segment PC. Connect AC. What is the maximum length of segment AC?"},{"type":"image_path","image_path":"images/10171_q0.png"}],"answer":"3$$\\sqrt{2}$$"}
{"id":"10216","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABCD, E is a point on side BC. Connect AE, and fold triangle ABE along AE; the corresponding point of B is F. If the extension of segment AF passes through the midpoint of one side of the rectangle, and AB = 2, AD = 4, then the length of BE is ______."},{"type":"image_path","image_path":"images/10216_q0.png"}],"answer":"$$2 \\sqrt{2} - 2$$ or $$\\frac{\\sqrt{17} - 1}{2}$$ or 2"}
{"id":"10227","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, square $$A B C D$$, point $$E$$ is a moving point on side $$A D$$, point $$A^{'}$$ is the reflection of point $$A$$ over line $$B E$$, connect $$A A^{'}$$ and extend it to intersect $$B E$$ at point $$G$$ and intersect $$C D$$ at point $$F$$, construct $$D H \\bot A F$$, given $$G H = 2$$, when point $$A^{'}$$, the reflection of point $$A$$, lies on diagonal $$B D$$, the value of $$\\frac{F D}{A D}$$ is        ; the area of square $$A B C D$$ is        ."},{"type":"image_path","image_path":"images/10227_q0.png"}],"answer":"$$\\sqrt{2} - 1$$     $$8 + 4 \\sqrt{2}$$"}
{"id":"10230","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A B = 8$$, $$\\angle B A D = 60^\\circ$$. Inside it, construct two rhombuses $$A E N H$$ and $$C G M F$$ of the same shape and size, such that points E, F, G, H lie on sides $$A B$$, $$B C$$, $$C D$$, $$D A$$ respectively, and points M, N lie on diagonal $$A C$$.\n(1) If $$A E = 3 B E$$, then the length of $$M N$$ is           .\n(2) If $$A E = B E$$, let P and Q be two moving points on $$D E$$ and $$A D$$ respectively, then the minimum value of $$A P + P Q$$ is           ."},{"type":"image_path","image_path":"images/10230_q0.png"}],"answer":"$$4 \\sqrt{3}$$     $$4 \\sqrt{3}$$"}
{"id":"10239","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$E$$ is the midpoint of $$A B$$, $$D$$ is a point on $$A C$$, connect $$D E$$, $$B H \\bot A C$$ at $$H$$, if $$2 \\angle A D E = 90 \\circ - \\angle H B C$$, $$A D : B C = 4 : 3$$, $$C D = 2$$, then the length of $$B C$$ is      ."},{"type":"image_path","image_path":"images/10239_q0.png"}],"answer":"6"}
{"id":"10240","difficulty":"0.4","question_list":[{"type":"text","text":"In △ABC, ∠A = 45°, ∠B = 60°, AB = 4, points P, M, N are on sides AB, BC, CA respectively, connect PM, MN, NP, then the minimum perimeter of △PMN is            ."},{"type":"image_path","image_path":"images/10240_q0.png"}],"answer":"$$2 \\sqrt{6}$$"}
{"id":"10251","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a square, point $$E$$ lies on the extension of $$C B$$, connect $$A E$$, $$A F \\bot A E$$ intersects $$C D$$ at point $$F$$, connect $$E F$$, point $$H$$ is the midpoint of $$E F$$, connect $$B H$$, then among the following conclusions: ① $$B E = D F$$; ② $$\\angle B E H = \\angle B A H$$; ③ $$\\frac{B H}{C F} = \\frac{\\sqrt{3}}{2}$$; ④ if $$A B = 4$$, $$D F = 1$$, then the area of $$\\triangle B E H$$ is $$\\frac{3}{4}$$. The correct ones are      . (Fill in all correct conclusion numbers on the line)"},{"type":"image_path","image_path":"images/10251_q0.png"}],"answer":"$$\\textcircled{1}\\textcircled{2}\\textcircled{4}$$"}
{"id":"10265","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$\\bigodot O$$ is the circumcircle of $$\\triangle A B C$$. Draw $$O D \\bot A B$$ through the center $$O$$, intersecting $$A B$$ at point $$D$$. Extend $$D O$$ to intersect $$\\bigodot O$$ at point $$F$$. Connect $$C F$$ and $$B F$$. Point $$C$$ is the midpoint of the minor arc $$B F$$. Take a point $$E$$ on the extension of $$A B$$, and connect $$C E$$, such that $$\\angle B C E = \\angle C A B$$. If $$B C = 10$$ and $$\\text{tan} \\angle C A B = \\frac{3}{4}$$, then the radius of $$\\bigodot O$$ is  , and $$\\frac{B E}{C E} =$$           ."},{"type":"image_path","image_path":"images/10265_q0.png"}],"answer":"$$\\frac{25}{3}$$     $$\\frac{25}{39}$$"}
{"id":"10266","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in △ABC, AB = AC = 4, BC = 2. Points P, E, F are arbitrary points on sides BC, AB, and AC, respectively. The minimum value of PE + PF is     ."},{"type":"image_path","image_path":"images/10266_q0.png"}],"answer":"$$\\frac{\\sqrt{15}}{2}$$"}
{"id":"10281","difficulty":"0.4","question_list":[{"type":"text","text":"In parallelogram $$A B C D$$, $$A C$$ is a diagonal, point $$E$$ is a point on $$A C$$, a line $$G F$$ passing through point $$E$$ intersects side $$C D$$ at point $$F$$ and side $$A B$$ at point $$G$$, connect $$D E$$. If $$D E \\bot G F$$, $$D F = 2 E F$$, $$\\angle A C B = 60 \\circ$$, $$\\frac{A C}{B C} = \\frac{5}{4}$$, then the value of $$\\frac{C F}{D F}$$ is             ."},{"type":"image_path","image_path":"images/10281_q0.png"}],"answer":"$$\\frac{1}{8}$$/$$0 . 125$$"}
{"id":"10302","difficulty":"0.4","question_list":[{"type":"text","text":"Given that the side length of square $$A B C D$$ is 4, and point $$P$$ is a point on ray $$B C$$, folding $$\\triangle A B P$$ along $$A P$$ yields $$\\triangle A E P$$. If $$\\triangle C D E$$ is an isosceles triangle, then the distance from point $$E$$ to $$B C$$ could be          ."},{"type":"image_path","image_path":"images/10302_q0.png"}],"answer":"$$4 - 2 \\sqrt{3}$$ or $$2$$ or $$4 + 2 \\sqrt{3}$$"}
{"id":"10305","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus $$A B C D$$, $$\\angle B = 60 \\circ$$, point $$E$$ is on side $$A B$$, with $$A E = 4$$, $$B E = 8$$, point $$F$$ is a point on $$B C$$, and $$\\triangle E G F$$ is a right triangle with right angle at point $$G$$ and $$\\angle E F G = 30^\\circ$$. Connect $$A G$$. When point $$F$$ moves along ray $$B C$$,\n(1) When $$B F = \\frac{1}{2} B E$$, $$A G =$$      .\n(2) The minimum value of segment $$D G$$ is      ."},{"type":"image_path","image_path":"images/10305_q0.png"}],"answer":"$$2 \\sqrt{7}$$     $$4 \\sqrt{3}$$"}
{"id":"10386","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the square paper $$ABCD$$ has a side length of 4. Point E is on side $$AD$$, and point F is on side $$CD$$. The square paper $$ABCD$$ is folded along EF, and the image of point B is point G. Connect $$DG$$. If $$AE = 1$$, then the minimum length of $$DG$$ is      ."},{"type":"image_path","image_path":"images/10386_q0.png"}],"answer":"$$\\sqrt{17} - 3$$"}
{"id":"10394","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the equilateral triangle $$\\triangle A B C$$ has side length 2. E and F are two moving points on sides $$B C$$ and $$C A$$, respectively, such that $$B E = C F$$. Connect $$A E$$ and $$B F$$, and let their intersection be P. Then the minimum value of $$P C$$ is          ."},{"type":"image_path","image_path":"images/10394_q0.png"}],"answer":"$$\\frac{2 \\sqrt{3}}{3}$$"}
{"id":"10422","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A C$$ is the diagonal, point F lies on $$A D$$, and segment $$B F$$ intersects $$A C$$ at point E, with $$\\angle A B F = \\angle F A C$$, and $$\\frac{A B}{B C} = \\frac{1}{2}$$;"},{"type":"image_path","image_path":"images/10422_q0.png"},{"type":"text","text":"(1) then $$\\frac{A F}{F D} =$$          ;  \n(2) If $$E G = \\sqrt{2}$$, $$\\triangle A B G$$ is an isosceles right triangle with $$A G = B G$$, then $$G H =$$           ."}],"answer":"$$\\frac{1}{3}$$     $$\\frac{5 \\sqrt{2}}{3}$$"}
{"id":"10431","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, point C lies on $$\\bigodot O$$, diameter $$AB = 25$$, $$CD \\bot AB$$, with foot at D, point E is the incenter of $$\\triangle BCD$$, $$AC = 15$$, point F lies on it, $$AF = 2FC$$, then $$EF =$$     ."},{"type":"image_path","image_path":"images/10431_q0.png"}],"answer":"$$\\sqrt{65}$$"}
{"id":"10433","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, draw ray CP on the right side of equilateral triangle ABC, $$\\angle A C P = \\alpha < 60^\\circ$$, point D is the reflection of point A over ray CP, BD intersects CP at point E, connect AD and AE. If $$A E = 3$$, $$C E = 4$$, then $$\\angle B E C =$$      , $$B D =$$         ."},{"type":"image_path","image_path":"images/10433_q0.png"}],"answer":"60°     10"}
{"id":"10456","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, the four corners of rectangle paper ABCD are folded inward, with EH, EF, FG, GH as the creases, where points A and B coincide at point J, and points C and D coincide at point K, and points H, J, K, F lie on the same straight line.\n(1) The shape of quadrilateral EFGH is            .\n(2) If $$\\frac{A H}{D H} = \\frac{3}{4}$$, JK = $$\\sqrt{2}$$, then AB =          ."},{"type":"image_path","image_path":"images/10456_q0.png"}],"answer":"rectangle;     $$4 \\sqrt{6}$$."}
{"id":"10457","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$\\square ABCD$$, $$AB = BC = 2$$, $$\\angle B = 120^\\circ$$, point $$E$$ is the reflection of point $$A$$ over line $$MN$$, and points $$N$$, $$E$$, $$M$$ lie on sides $$AB$$, $$BC$$, $$AD$$ respectively. Then the maximum value of $$MD$$ is      ."},{"type":"image_path","image_path":"images/10457_q0.png"}],"answer":"$$2 - \\sqrt{3}$$/$$- \\sqrt{3} + 2$$"}
{"id":"10459","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = B C$$, point D is any point on side BC, connect AD, from point C draw $$C E \\bot A D$$ at point E, from point C draw $$C F \\bot C E$$, and $$C F = C E$$, connect FE and extend it to intersect AB at point M, connect BF. If the area of quadrilateral AMEC is 8, $$C E = 2$$, then the area of quadrilateral ABFC is        ."},{"type":"image_path","image_path":"images/10459_q0.png"}],"answer":"18"}
{"id":"10470","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a rhombus, with diagonals $$A C$$, $$B D$$ intersecting at point O, $$A C = 2 \\sqrt{3}$$, $$B D = 2$$, point P is a moving point on $$A C$$, and point E is the midpoint of $$A B$$. Then the minimum value of $$P D + P E$$ is        ."},{"type":"image_path","image_path":"images/10470_q0.png"}],"answer":"$$\\sqrt{3}$$"}
{"id":"10483","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, place a set of triangle rulers as shown, with point E on side AC. Rotate triangle $$\\triangle ABC$$ around point A at a speed of 3° per second in a clockwise direction for one full rotation. During the rotation, at the     second, side BC is exactly parallel to side DE."},{"type":"image_path","image_path":"images/10483_q0.png"}],"answer":"35 or 95"}
{"id":"10504","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, point $$D$$ lies on $$B C$$, connect $$A D$$, $$\\angle D A C = 45 \\circ$$, point $$E$$ lies on $$A D$$, connect $$C E$$, $$\\angle B C E = 45 \\circ$$, if $$C E = B C$$, and the area of $$\\triangle A B C$$ is $$8 \\sqrt{2}$$, then the length of $$A C$$ is    ."},{"type":"image_path","image_path":"images/10504_q0.png"}],"answer":"$$4 \\sqrt{2}$$"}
{"id":"10510","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$R t \\triangle A B C$$, $$\\angle C = 90 \\circ$$, $$A C = 3$$, $$B C = 4$$, points $$D$$, $$E$$ are the midpoints of sides $$C A$$, $$C B$$ respectively, the angle bisector of $$\\angle C A B$$ intersects $$D E$$ at point $$F$$, then the length of $$C F$$ is          ."},{"type":"image_path","image_path":"images/10510_q0.png"}],"answer":"$$\\frac{3 \\sqrt{5}}{5}$$"}
{"id":"10520","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the triangular paper $$\\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A B = 10$$, $$B C > A C$$, points D and E lie on AB. Fold $$\\angle A$$ along CD so that point A lands at point $$A^{'}$$ on AB; fold $$\\angle B$$ along CE so that point B lands on the extension of $$C A^{'}$$, with B's corresponding point being $$B^{'}$$. If the area of $$\\triangle A^{'} E B^{'}$$ is $$\\frac{9}{4}$$, then the value of $$A C \\cdot B C$$ is        ."},{"type":"image_path","image_path":"images/10520_q0.png"}],"answer":"5 or 45"}
{"id":"10526","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle ABMN, AN = $$\\sqrt{2}$$, point C is the midpoint of MN, connect AC and BC, and BC = 2$$\\sqrt{2}$$, point D is the midpoint of AC, point E is a moving point on side AB, connect DE, point F is the reflection of point A over line DE, connect DF and EF. When EF ⊥ AC, the length of AE is    ."},{"type":"image_path","image_path":"images/10526_q0.png"}],"answer":"$$\\frac{\\sqrt{6}}{3}$$ or $$\\sqrt{6}$$"}
{"id":"10572","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram $$A B C D$$, $$A B = 4$$, $$B C = 6$$, $$\\angle A B C = 60 \\circ$$, point P is a point in the plane satisfying $$\\angle A P D = 90 \\circ$$, where point Q is a point on segment $$C P$$ such that $$\\frac{P Q}{C Q} = \\frac{1}{2}$$, connect $$B Q$$, then the minimum value of $$B Q$$ is         ."},{"type":"image_path","image_path":"images/10572_q0.png"}],"answer":"$$\\frac{4 \\sqrt{19}}{3} - 2$$"}
{"id":"10579","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, students from the school labor club used mathematical knowledge to sketch the emblem of the club. The design process is as follows: Construct an isosceles triangle $$\\triangle ABC$$ inscribed in circle $$O$$, draw $$AD \\bot BC$$ intersecting $$BC$$ at point $$E$$ and intersecting circle $$O$$ at point $$D$$, draw $$EF \\bot AB$$ at point $$F$$ and $$EG \\bot AC$$ at point $$G$$, connect $$DF$$ and $$DG$$. It is measured that $$DE = 2 \\text{cm}$$, $$BC = 8 \\text{cm}$$. The area of quadrilateral $$AFDG$$ is      $$\\text{cm}^{\\text{2}}$$."},{"type":"image_path","image_path":"images/10579_q0.png"}],"answer":"$$32$$"}
{"id":"10617","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rhombus $$ABCD$$ with side length 4, $$\\angle BAD = 60^\\circ$$. There is a triangle board $$\\triangle BEF$$ with $$\\angle BFE = 30^\\circ$$. When $$\\triangle BEF$$ is rotated about point $$B$$ to form $$\\triangle BE'F'$$, the lines containing $$BE'$$ and $$BF'$$ intersect segment $$AC$$ at points $$M$$ and $$N$$, respectively. If $$C'$$ is the reflection of point $$C$$ across line $$BE'$$, and when $$C'N \\perp AC$$, the length of $$AN$$ is           ."},{"type":"image_path","image_path":"images/10617_q0.png"}],"answer":"$$2 \\sqrt{3} - 2$$/$$- 2 + 2 \\sqrt{3}$$"}
{"id":"10633","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle B A C = 52^\\circ$$, $$A B = A C$$, $$B D \\bot A C$$ at D, E and F move along segments $$B D$$ and $$B C$$ respectively, and $$B E = C F$$. When $$A E + A F$$ is minimized, the measure of $$\\angle A E D$$ is          ."},{"type":"image_path","image_path":"images/10633_q0.png"}],"answer":"$$77 \\circ$$"}
{"id":"10637","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$ABCD$$, moving points $$E$$ and $$F$$ start simultaneously from points $$D$$ and $$C$$, respectively, moving at the same speed along sides $$DC$$ and $$CB$$, respectively. Connect $$AE$$ and $$DF$$, intersecting at point $$P$$. As points $$E$$ and $$F$$ move, point $$P$$ also moves accordingly. If $$AD = 2$$, the minimum value of segment $$CP$$ is          ."},{"type":"image_path","image_path":"images/10637_q0.png"}],"answer":"$$\\sqrt{\\text{5}} −\\text{1}$$"}
{"id":"10653","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A C$$ and $$B D$$ intersect at point $$O$$, the perpendicular bisector of side $$A B$$ intersects $$B D$$ at point $$E$$ and intersects $$A B$$ at point $$F$$, point $$G$$ is the midpoint of side $$C D$$, connect $$E G$$, if $$A C = 8$$, $$B D = 16$$, then the length of segment $$E G$$ is      ."},{"type":"image_path","image_path":"images/10653_q0.png"}],"answer":"$$\\sqrt{53}$$"}
{"id":"10655","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in the acute triangle $$\\triangle A B C$$, $$\\angle B A C = 45 \\circ$$, and $$A D \\bot B C$$ at point D. If $$B D = a$$, $$A D = b$$, then the length of $$C D$$ is      . (Expressed as an algebraic expression in terms of a and b)"},{"type":"image_path","image_path":"images/10655_q0.png"}],"answer":"$$\\frac{b^{2} - a b}{a + b}$$"}
{"id":"10659","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta A B C$$, $$\\angle B = 90 \\circ$$, construct the angle bisectors of the interior angle $$\\angle A C B$$ and the exterior angle $$\\angle D A C$$, and let the lines containing these two angle bisectors intersect at point $$E$$, then $$\\angle E =$$    degrees; construct the angle bisectors of $$\\angle E A B$$ and $$\\angle E C B$$, and let these two angle bisectors intersect at point $$F$$, then $$\\angle A F C =$$      degrees."},{"type":"image_path","image_path":"images/10659_q0.png"}],"answer":"45     67.5"}
{"id":"10661","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, ∠AOB = 50°, ∠BOC = 30°, OM = 11, ON = 6. Points P and Q are moving points on OA and OB respectively. The minimum value of MQ + PQ + NP is    ."},{"type":"image_path","image_path":"images/10661_q0.png"}],"answer":"$$\\sqrt{223}$$"}
{"id":"10681","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, it is given that quadrilateral $$A B C D$$ is a square with all four angles right angles and all four sides equal. Point $$E$$ lies on $$B C$$ such that $$C E = \\frac{1}{4} B C$$, and point $$F$$ is the midpoint of $$C D$$. Extend $$A F$$ to intersect the extension of $$B C$$ at point $$M$$. The following conclusions: ① $$A B = C M$$; ② $$A E = A B + C E$$; ③ $$S_{\\triangle A E F} = \\frac{1}{4} S_{A B C F}$$; ④ $$\\angle A F E = 90 \\circ$$, among which the correct conclusions are      ."},{"type":"image_path","image_path":"images/10681_q0.png"}],"answer":"①②④"}
{"id":"10704","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = 3$$, $$B C = 4$$, $$A B = 5$$. Quadrilateral $$A B E F$$ is a square. Point D is a point on line $$B C$$ such that $$C D = 1$$. Point P is a point on segment $$D E$$ such that $$P D = \\frac{2}{3} D E$$. Draw line a through point P parallel to $$B C$$, intersecting segment $$A B$$ at point G and segment $$A D$$ at point H. Then the length of $$G H$$ is           ."},{"type":"image_path","image_path":"images/10704_q0.png"}],"answer":"$$\\frac{1}{3}$$ or $$\\frac{5}{9}$$/$$\\frac{5}{9}$$ or $$\\frac{1}{3}$$"}
{"id":"10706","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in square $$A B C D$$, point $$E$$ is a point on $$C D$$, and point $$F$$ is a point on the extension of $$C B$$, with $$D E = B F$$. Connect $$B D$$ and $$E F$$, intersecting at point $$G$$, and connect $$A G$$. Among the following conclusions, ① $$\\triangle A E F$$ is an isosceles right triangle; ② $$A G \\bot E F$$; ③ $$C E = 2 B G$$; ④ $$A B - B F = \\sqrt{2} B G$$; ⑤ when $$E C = 3 D E$$, $$\\text{sin} \\angle D G E = \\frac{1}{4}$$. The correct ones are          ."},{"type":"image_path","image_path":"images/10706_q0.png"}],"answer":"①②④"}
{"id":"10731","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, rays $$AM$$, $$BN$$ are both perpendicular to segment $$AB$$. From point A, draw perpendicular $$AC$$ to $$BE$$, intersecting $$BE$$ and $$BN$$ at points F and C respectively, with foot at D. If $$CD = CF$$, then $$\\frac{AE}{AD} =$$                    ."},{"type":"image_path","image_path":"images/10731_q0.png"}],"answer":"$$\\frac{\\sqrt{2} - 1}{2}$$"}
{"id":"10741","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$O$$ is the intersection point of the diagonals $$AC$$ and $$BD$$ of square $$ABCD$$, and $$E$$, $$F$$ are points on sides $$AB$$ and $$BC$$ respectively, such that $$BE + EF = FC$$; then $$\\angle EOF =$$      degrees; if $$OE = \\frac{\\sqrt{6}}{2} OF$$, then $$\\frac{AE}{CF} =$$      ."},{"type":"image_path","image_path":"images/10741_q0.png"}],"answer":"45     $$\\frac{3}{2}$$"}
{"id":"10765","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, given that the side length of square $$A B C D$$ is 2, point $$E$$ is the midpoint of side $$C D$$, and $$\\triangle A D E$$ is folded along $$A E$$ to $$\\triangle A F E$$, extending $$A F$$ intersects side $$B C$$ at point $$G$$, then $$C G =$$      ; if extending $$E F$$ intersects side $$B C$$ at point $$Q$$, then $$t a n \\angle E Q C =$$      ."},{"type":"image_path","image_path":"images/10765_q0.png"}],"answer":"$$\\frac{1}{2}$$/$$0.5$$     $$\\frac{3}{4}$$/$$0.75$$"}
{"id":"10785","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, $$C$$, $$D$$ are the trisection points of $$AB$$, arcs are drawn with centers at $$C$$ and $$D$$ and radius equal to $$CD$$, intersecting at points $$E$$ and $$F$$, and $$EF$$ is connected. If $$AB = 9$$, then the length of $$EF$$ is        ."},{"type":"image_path","image_path":"images/10785_q0.png"}],"answer":"$$3 \\sqrt{3}$$"}
{"id":"10786","difficulty":"0.4","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 3$$, point $$E$$ is any point on side $$C D$$ (point $$E$$ does not coincide with points $$C$$ and $$D$$). From point $$A$$, draw $$A F \\bot A E$$, intersecting the extension of side $$C B$$ at point $$F$$. Connect $$E F$$, intersecting side $$A B$$ at point $$G$$. Let $$D E = x$$, $$B F = y$$. Then the function expression of $$y$$ in terms of $$x$$ is      ; when point $$E$$ moves along side $$C D$$, if $$\\triangle A E G$$ is an isosceles triangle, the length of segment $$D E$$ is      ."},{"type":"image_path","image_path":"images/10786_q0.png"}],"answer":"$$y = \\frac{4}{3} x \\left(0 < x < 4\\right)$$     $$\\frac{9}{4}$$ or $$\\frac{3}{2}$$ or $$\\frac{7}{8}$$"}
{"id":"10799","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠ACB = 90°, AC = 5 cm, BC = 12 cm. Rotate triangle ABC clockwise around point B by 60° to obtain triangle BDE. Connect DC and let it intersect AB at point F. The sum of the perimeters of triangles ACF and BDF is ______ cm."},{"type":"image_path","image_path":"images/10799_q0.png"}],"answer":"42"}
{"id":"10802","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A B = 4$$, a circle $$\\bigodot O$$ is drawn with $$A B$$ as its diameter, intersecting $$B C$$ at point E. A tangent line $$A F$$ to $$\\bigodot O$$ is drawn through point A, intersecting $$C D$$ at point F. Connect $$B F$$ and $$A E$$. If $$\\angle E A F = 60^\\circ$$, then the length of $$B F$$ is       ."},{"type":"image_path","image_path":"images/10802_q0.png"}],"answer":"$$2 \\sqrt{7}$$"}
{"id":"10807","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in trapezoid $$A B C D$$, $$A D \\parallel B C$$, $$\\angle A = 90 \\circ$$, $$A D = 3$$, $$A B = 4$$, $$B C = 6$$. Point E lies on side $$B C$$, and triangle $$A B E$$ is folded along $$A E$$, with point B mapping to point F. If $$E F \\bot C D$$, then the length of $$E C$$ is      ."},{"type":"image_path","image_path":"images/10807_q0.png"}],"answer":"$$\\frac{14}{3}$$"}
{"id":"10819","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A B = 20$$, $$A C = 16$$, $$B C = 12$$, a semicircle is drawn with the midpoint O of side AB as its center, tangent to AC. Points P and Q are moving points on side BC and the semicircle, respectively. Connect PQ; the minimum length of PQ is      ."},{"type":"image_path","image_path":"images/10819_q0.png"}],"answer":"2"}
{"id":"10832","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle A B C = 90 \\circ$$, $$B D$$ is the median of $$\\triangle A B C$$, $$D E \\bot B C$$ at point E. If $$B E = 3$$, $$D E = 4$$, then the area of quadrilateral $$A B E D$$ is         ."},{"type":"image_path","image_path":"images/10832_q0.png"}],"answer":"18"}
{"id":"10841","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A C D$$ and $$\\triangle B D C$$, $$\\angle A = \\angle B = 90 \\circ$$, $$A D = B C$$, $$\\angle A C D = 35 \\circ$$, then $$\\angle A C B =$$        $$\\circ$$."},{"type":"image_path","image_path":"images/10841_q0.png"}],"answer":"$$20$$"}
{"id":"10857","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$B C = 16$$. Fold rectangle $$A B C D$$ along $$E F$$ so that point C coincides with point A. Then the length of the crease $$E F$$ is     ."},{"type":"image_path","image_path":"images/10857_q0.png"}],"answer":"$$4 \\sqrt{5}$$"}
{"id":"10863","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in right triangle ABC, ∠C = 90°, ∠BAC = 60°, point D is a point on side BC. Connect AD, fold triangle ACD along AD so that point C lands at point E. When triangle BDE is a right triangle, the measure of ∠CAD is           ."},{"type":"image_path","image_path":"images/10863_q0.png"}],"answer":"$$30 \\circ$$ or $$45 \\circ$$"}
{"id":"10895","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$D E$$ is the perpendicular bisector of $$B C$$, intersecting $$A B$$ at point D and $$B C$$ at point E. If $$A B = 8$$, $$A C = 6$$, then the perimeter of $$\\triangle A C D$$ is      ."},{"type":"image_path","image_path":"images/10895_q0.png"}],"answer":"14"}
{"id":"10967","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, connect $$A C$$, $$\\angle D A B$$ and $$\\angle B$$ are complementary, $$P M$$ and $$P N$$ are the midlines of $$\\triangle A D C$$ and $$\\triangle C B A$$ respectively, connect $$M N$$, $$P M = P N$$, if $$A B = 5$$, $$C D = 2$$, then the minimum value of segment $$A D$$ is      ."},{"type":"image_path","image_path":"images/10967_q0.png"}],"answer":"$$\\frac{3 \\sqrt{2}}{2}$$/$$\\frac{3}{2} \\sqrt{2}$$"}
{"id":"10968","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral $$A B C D$$ is a rectangle. Connect $$A C$$. Points $$E$$ and $$F$$ are the midpoints of sides $$A B$$ and $$A D$$, respectively. Connect $$E F$$, $$E F = \\sqrt{5}$$. $$A M \\bot A C$$ intersects the extension of $$C B$$ at point $$M$$. Point $$N$$ is the midpoint of $$C M$$. Connect $$A N$$. If $$t a n \\angle A M C = 2$$, then $$A N =$$"},{"type":"image_path","image_path":"images/10968_q0.png"}],"answer":"$$\\frac{5}{2}$$"}
{"id":"10977","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, quadrilateral ABCD is a rhombus with AB = 8 and ∠ABC = 60°. M is any point on diagonal BD (excluding point B). The minimum value of AM + $$\\frac{1}{2}$$BM is     ."},{"type":"image_path","image_path":"images/10977_q0.png"}],"answer":"4$$\\sqrt{3}$$"}
{"id":"10990","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the cross-section of a certain dam is a trapezoid $$ABCD$$, with $$AE$$ and $$DF$$ as the heights of the trapezoid. The slope angle of the upstream slope $$AB$$ is $$\\alpha = 45^\\circ$$, and the slope length $$AB = 6\\sqrt{2}$$ m. The slope ratio of the downstream slope $$CD$$ is $$i = 1 : \\sqrt{3}$$. Then the length of the downstream slope is      m."},{"type":"image_path","image_path":"images/10990_q0.png"}],"answer":"12"}
{"id":"10991","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$A C = B C$$. With point C as the center and an appropriate length as the radius, draw arcs intersecting $$A C$$ and $$B C$$ at points E and F, respectively. Then, with points E and F as centers and a radius greater than $$\\frac{1}{2} E F$$, draw arcs intersecting at point G. Draw ray $$C G$$ intersecting $$A B$$ at point D. Through point D, draw $$D H \\parallel B C$$ intersecting $$A C$$ at point H. If $$C H = a$$, then $$B C =$$      (expressed as an algebraic expression in terms of a)."},{"type":"image_path","image_path":"images/10991_q0.png"}],"answer":"$$2 a$$"}
{"id":"10992","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$, chord $$MN \\parallel AB$$, perpendiculars are drawn from $$M$$ and $$N$$ to $$AB$$, with feet at $$C$$ and $$D$$, respectively. The following conclusions"},{"type":"image_path","image_path":"images/10992_q0.png"},{"type":"text","text":"① $$AC = BD$$;  \n② $$\\overset{⌢}{AM} = \\overset{⌢}{BN}$$;  \n③ If quadrilateral $$MC DN$$ is a square, then $$MN = \\frac{1}{2} AB$$;  \n④ If $$M$$ is the midpoint of arc $$AN$$, then $$D$$ is the midpoint of $$OB$$.  \nThe sequence numbers of all correct conclusions are      ."}],"answer":"①②④"}
{"id":"10994","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\triangle A B C$$, $$\\angle C A B = 60 \\circ$$. With $$A$$ as the center, draw an arc of appropriate length as the radius, intersecting $$A B$$ and $$A C$$ at points $$D$$ and $$E$$, respectively. Then, with $$D$$ and $$E$$ as centers, draw arcs with radii greater than $$\\frac{1}{2} D E$$, intersecting at point $$F$$. Draw ray $$A F$$, intersecting $$B C$$ at point $$P$$. Through point $$P$$, draw lines parallel to $$A C$$ and $$A B$$, intersecting $$A B$$ and $$A C$$ at points $$M$$ and $$N$$, respectively. If $$A P = 4 \\sqrt{3}$$, then the area of quadrilateral $$A M P N$$ is      ."},{"type":"image_path","image_path":"images/10994_q0.png"}],"answer":"$$8 \\sqrt{3}$$"}
{"id":"10995","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\bigodot O$$, point $$C$$ is the midpoint of $$\\overset{⌢}{A B}$$, and connecting $$O C$$ intersects chord $$A B$$ at point $$D$$. If $$O D = 3$$ and $$D C = 2$$, then the length of $$A B$$ is      ."},{"type":"image_path","image_path":"images/10995_q0.png"}],"answer":"8."}
{"id":"10999","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, rectangle $$A B C D$$, point E is a point on $$A B$$, connect $$C E$$, take a point F on $$C E$$, connect $$B F$$, draw a perpendicular from F to $$C E$$ intersecting $$A D$$ at point H. If $$\\angle E B F + \\angle B C E = 90 \\circ$$, $$C E = 2 F H$$, $$A D = 6$$, $$C E = 2 \\sqrt{10}$$, then the length of $$C D$$ is           ."},{"type":"image_path","image_path":"images/10999_q0.png"}],"answer":"4"}
{"id":"11006","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in sector $$A O B$$, $$\\angle A O B = 120 \\circ$$, $$A O = 6$$, point $$C$$ is the midpoint of arc $$A B$$, connect $$O C$$, intersecting $$A B$$ at point $$D$$, $$E$$ is the midpoint of $$O D$$, connect $$B E$$, then the area of the shaded region in the figure is       . (Express the result in terms of radicals and $$\\pi$$)"},{"type":"image_path","image_path":"images/11006_q0.png"}],"answer":"$$6 \\pi - \\frac{9}{4} \\sqrt{3}$$"}
{"id":"11007","difficulty":"0.6","question_list":[{"type":"text","text":"The ancient Greek mathematician Euclid, in his deep study of proportion theory, proposed the \"extreme and mean ratio\" problem of dividing a line segment: point G divides a line segment into two parts such that the longer segment is the geometric mean of the entire length and the shorter segment, i.e., it satisfies $$\\frac{MG}{MN} = \\frac{GN}{MG} = \\frac{\\sqrt{5} - 1}{2}$$. Later generations called the number $$\\frac{\\sqrt{5} - 1}{2}$$ the \"golden ratio\" and referred to point G as the \"golden ratio\" point of the segment. As shown in the figure, in △ABC, given AB = AC = 3, BC = 4, if D and E are two \"golden ratio\" points on the sides, then the area of △ADE is      ."},{"type":"image_path","image_path":"images/11007_q0.png"}],"answer":"10-4$$\\sqrt{5}$$"}
{"id":"11020","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the equilateral triangle △ABC, take a point P such that the lengths of PA, PB, and PC are m, m, and $$\\sqrt{2} m$$, respectively. Rotate segment BP clockwise 60° about point B to obtain segment BQ, and connect CQ. Then the measure of $$\\angle A P B$$ is      ."},{"type":"image_path","image_path":"images/11020_q0.png"}],"answer":"$$150 \\circ$$"}
{"id":"11024","difficulty":"0.6","question_list":[{"type":"text","text":"In rectangle $$A B C D$$, $$A B = 6$$, $$A D = 8$$, $$P$$ is a moving point on side $$C D$$, connect $$A P$$, draw $$P M \\bot A P$$, on $$A P$$ take $$P M = \\frac{3}{4} P N$$, draw $$P H \\bot M N$$ at $$H$$, connect $$D H$$, then the minimum value of $$D H$$ is       ."},{"type":"image_path","image_path":"images/11024_q0.png"}],"answer":"$$\\frac{24}{5}$$"}
{"id":"11028","difficulty":"0.6","question_list":[{"type":"text","text":"The data for the three sides of the triangle are shown in the figure; then the range of $$x$$ is           ."},{"type":"image_path","image_path":"images/11028_q0.png"}],"answer":"$$- 2 < x < 0$$"}
{"id":"11029","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $AC$ and $BD$ of square $ABCD$ intersect at point $O$. Point $E$ is a point on $AD$. Connect $EO$ and extend it to intersect $BC$ at point $F$. If $\\angle DOE = 22.5^{\\circ}$ and $BF = 2 - \\sqrt{2}$, then the perimeter of square $ABCD$ is        ."},{"type":"image_path","image_path":"images/11029_q0.png"}],"answer":"$$8$$"}
{"id":"11034","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$, point $$P$$ is a moving point inside rectangle $$A B C D$$, and $$S_{\\Delta P A B} = S_{\\Delta P C D}$$, then the minimum value of $$P C + P D$$ is     ."},{"type":"image_path","image_path":"images/11034_q0.png"}],"answer":"$$2 \\sqrt{13}$$"}
{"id":"11054","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the diagonals of rhombus $$A B C D$$ intersect at point $$O$$. If $$A C = 24$$, $$B D = 10$$, then the perimeter of rhombus $$A B C D$$ is          ."},{"type":"image_path","image_path":"images/11054_q0.png"}],"answer":"$$52$$"}
{"id":"11063","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the rhombus $$A B C D$$ has side length 4, $$\\angle A = 60 \\circ$$, and when folded along $$E F$$, vertex C lands exactly at the midpoint G of side $$A B$$. Then $$B F =$$        ."},{"type":"image_path","image_path":"images/11063_q0.png"}],"answer":"$$1.2$$"}
{"id":"11068","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\text{Rt} \\triangle A B C$$, $$\\angle A C B = 90 \\circ$$, $$A C = 8$$, $$B C = 6$$, point $$D$$ is a moving point on side $$A B$$, and $$\\triangle A C D$$ is folded along side $$C D$$ to obtain $$\\triangle C D E$$. When the overlapping part of $$\\triangle C D E$$ and $$\\triangle A B C$$ is a right triangle, the length of $$A D$$ is       ."},{"type":"image_path","image_path":"images/11068_q0.png"}],"answer":"4 or $$\\frac{32}{5}$$"}
{"id":"11092","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in $$\\Delta A B C$$, $$s i n B = \\frac{1}{4}$$, $$t a n C = \\frac{1}{2}$$, $$A B = 4$$, then the length of $$A C$$ is        ."},{"type":"image_path","image_path":"images/11092_q0.png"}],"answer":"$$\\sqrt{5}$$"}
{"id":"11119","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in rectangle $$A B C D$$, $$A B = 3 \\text{cm}$$, $$B C = 4 \\text{cm}$$. The rectangle is folded so that point $$C$$ coincides with point $$A$$. The length of the crease is       ."},{"type":"image_path","image_path":"images/11119_q0.png"}],"answer":"$$\\frac{15}{4}$$"}
{"id":"11121","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, $$AB$$ is the diameter of $$\\bigodot O$$. $$BD$$ bisects $$\\angle ABC$$, $$DB$$ intersects $$AC$$ at point $$E$$, $$\\angle DBA = 30^\\circ$$. If $$BC = 2$$, then the area of $$\\triangle BCE$$ is      ."},{"type":"image_path","image_path":"images/11121_q0.png"}],"answer":"$$\\frac{2 \\sqrt{3}}{3}$$"}
{"id":"11127","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, the diagonals $$AC$$ and $$BD$$ of rhombus $$ABCD$$ intersect at point O, $$BE \\parallel AC$$, $$AE \\parallel BD$$, and $$OE$$ intersects $$AB$$ at point F. If $$OE = 5$$ and $$AC = 8$$, then the area of rhombus $$ABCD$$ is        ."},{"type":"image_path","image_path":"images/11127_q0.png"}],"answer":"24"}
{"id":"11129","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$A B = 5$$, $$A C = 8$$, point $$P$$ is a moving point on side $$B C$$, and triangle $$\\triangle A B P$$ is folded along $$A P$$ such that vertex $$B$$ lands at point $$B^{'}$$ inside the rhombus $$A B C D$$. When points $$P$$, $$B^{'}$$, and $$D$$ are collinear, the distance from point $$A$$ to line $$P D$$ is      ."},{"type":"image_path","image_path":"images/11129_q0.png"}],"answer":"$$\\frac{24}{5}$$"}
{"id":"11130","difficulty":"0.6","question_list":[{"type":"text","text":"Given that OA is the radius of ⊙O, extend AO to point B such that OB = 3OA = 3. Construct an isosceles right triangle BMC with right angle at B, such that point M always lies on ⊙O (as shown in the figure). Connect OC; then the maximum value of OC is      ."},{"type":"image_path","image_path":"images/11130_q0.png"}],"answer":"$$3 \\sqrt{2} + 1$$/$$1 + 3 \\sqrt{2}$$"}
{"id":"11156","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, AB = 3, AD = 4, ∠ABC = 60°. Through the midpoint E of BC, draw EF ⊥ AB, with foot at point F, intersecting the extension of DC at point H. The area of triangle DEF is     ."},{"type":"image_path","image_path":"images/11156_q0.png"}],"answer":"2$$\\sqrt{3}$$"}
{"id":"11167","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in parallelogram ABCD, a line passing through the midpoint O of diagonal AC intersects BC and AD at points E and F, respectively. Only one additional condition is needed to prove that quadrilateral AECF is a rectangle; this condition can be        (write one such condition)."},{"type":"image_path","image_path":"images/11167_q0.png"}],"answer":"AC = FE or AE ⟂ BC, etc. (The answer is not unique; any condition that satisfies the problem statement is acceptable.)"}
{"id":"11173","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ bisects $$\\angle B A D$$, $$B C = C D = 2$$, $$A B = 5$$, $$A D = 3$$, then the length of $$A C$$ is       ."},{"type":"image_path","image_path":"images/11173_q0.png"}],"answer":"$$\\sqrt{19}$$"}
{"id":"11175","difficulty":"0.6","question_list":[{"type":"text","text":"In rectangle $$A B C D$$, $$A B = 3$$, $$A D = 3 \\sqrt{3}$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, from point $$A$$ draw $$A E \\bot B O$$, with foot at $$E$$, $$N$$ is the midpoint of $$A D$$, connect $$B N$$ intersecting $$A E$$ at point $$P$$, then the length of $$P E$$ is        ."},{"type":"image_path","image_path":"images/11175_q0.png"}],"answer":"$$\\frac{3 \\sqrt{3}}{10}$$/$$\\frac{3}{10} \\sqrt{3}$$"}
{"id":"11182","difficulty":"0.6","question_list":[{"type":"text","text":"As shown in the figure, in the rhombus $$A B C D$$, $$\\angle A B C = 120 \\circ$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$B D = 8$$, point $$E$$ is the midpoint of $$O D$$, point $$F$$ is a point on $$A B$$ such that $$A F = 3 B F$$, point $$P$$ is a moving point on $$A C$$, connect $$P E$$, $$P F$$, then the maximum value of $$\\left|P F - P E\\right|$$ is        ."},{"type":"image_path","image_path":"images/11182_q0.png"}],"answer":"$$2$$"}