Datasets:
id string | difficulty string | question_list list | answer string |
|---|---|---|---|
3 | 0.2 | [
{
"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.... | ①②③④⑤ |
6 | 0.2 | [
{
"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 \"be... | $$36 \circ$$ |
11 | 0.2 | [
{
"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 ."
... | $$\sqrt{21}$$ |
18 | 0.2 | [
{
"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... | $$\sqrt{13}$$ $$1 + \frac{3 \sqrt{3}}{2}$$ |
20 | 0.2 | [
{
"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"
}
] | $$\sqrt{5} - 1$$ |
30 | 0.2 | [
{
"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 $$P B^{'}$$ intersec... | (1)(2)(4) |
31 | 0.2 | [
{
"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 $$A B = B E$$, $$\\angle D A C = 2 ... | $$2 \sqrt{10}$$ |
32 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ to obtain quadrila... | $$\frac{15 + 9 \sqrt{5}}{5}$$ |
35 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ at poin... | $$\frac{13}{18}$$ or $$\frac{5}{7}$$ or $$\frac{10}{27}$$ |
42 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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^{'}$$, draw $$... | $$\frac{200}{39}$$ |
46 | 0.2 | [
{
"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 $$A$... | $$1$$ or $$2$$ or $$\sqrt{2}$$. |
47 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the square $$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$... | $$\frac{7}{9} \sqrt{10}$$ |
49 | 0.2 | [
{
"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 $$\\bigodot... | 9 $$\frac{12 \sqrt{7}}{7}$$ |
62 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, it is given that $$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\\sq... | $$\frac{14}{3}$$ |
78 | 0.2 | [
{
"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 $$MN$$... | $$\frac{\sqrt{5}}{3}$$/$$\frac{1}{3} \sqrt{5}$$ |
85 | 0.2 | [
{
"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... | $$\frac{6}{5} \sqrt{5}$$ |
86 | 0.2 | [
{
"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 $$\\triangle A^{'} B^{' '} C^{'... | $$2 \sqrt{7} + 2$$ |
87 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the side length of square $$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$$, intersecting at point... | $$\frac{2 \sqrt{5}}{5}$$ 1 |
94 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure: In quadrilateral $$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{... | $$\frac{3}{5}$$ or $$\frac{12}{13}$$ |
108 | 0.2 | [
{
"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 $$B E = A F$$. $$B F$$... | ①②④ |
111 | 0.2 | [
{
"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"... | $$2 \sqrt{3}$$ |
115 | 0.2 | [
{
"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 $$P$$ s... | $$\frac{3 \sqrt{3}}{2}$$ or $$\frac{17 \sqrt{123} - 99 \sqrt{3}}{16}$$ or $$\frac{4 \sqrt{141} - 12}{11}$$ |
117 | 0.2 | [
{
"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 $$D ... | $$\sqrt{7}$$ |
118 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$,... | $$\frac{2 \sqrt{31}}{3}$$ |
124 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, $$\\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$$.... | $$3 \sqrt{3}$$ |
126 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, $$\\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$$ (with points B and C corresponding to points D ... | $$4 - \sqrt{2}$$/$$- \sqrt{2} + 4$$ |
129 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$ is ... | 7.5/$$\frac{15}{2}$$ |
130 | 0.2 | [
{
"type": "text",
"text": "Given in $$\\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$$, intersecting $$... | $$\frac{10}{3}$$/$$3 \frac{1}{3}$$ |
132 | 0.2 | [
{
"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. ... | ①②④ |
136 | 0.2 | [
{
"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 $... | $$2 \sqrt{3}$$ |
140 | 0.2 | [
{
"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 $$C D = 4 \\sqrt{... | $$\textcircled{1} \textcircled{2} \textcircled{4}$$ |
141 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in parallelogram $$\\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}... | $$\frac{20 \sqrt{5}}{19}$$/$$\frac{20}{19} \sqrt{5}$$ |
142 | 0.2 | [
{
"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}\textcircled{3}\textcircled{4}$$ |
145 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, given that the side length of square $$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$$, then the minimum valu... | $$8 \sqrt{5}$$ |
151 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, $$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$... | $$\pi$$ |
153 | 0.2 | [
{
"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... | $$3 \sqrt{5} - 3 \sqrt{2}$$ |
157 | 0.2 | [
{
"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... | 2 |
159 | 0.2 | [
{
"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 $$O D^{'}... | $$\sqrt{2}$$ |
170 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in square $$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 ... | ①②③ |
179 | 0.2 | [
{
"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",
... | 12 |
187 | 0.2 | [
{
"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 $... | $$\frac{1}{3}$$ |
191 | 0.2 | [
{
"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 ... | $$9 - 3 \sqrt{3}$$/$$- 3 \sqrt{3} + 9$$ |
193 | 0.2 | [
{
"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 $$C... | $$3 \sqrt{3}$$ |
207 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in the rhombus $$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$$ is a ... | $$\frac{8 \sqrt{2}}{3}$$/$$\frac{8}{3} \sqrt{2}$$ |
212 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the side length of square $$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 = ... | $$\frac{9}{8}$$ or $$\frac{25}{8}$$ |
213 | 0.2 | [
{
"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... | $$\sqrt{3} - \sqrt{2}$$/$$- \sqrt{2} + \sqrt{3}$$ |
216 | 0.2 | [
{
"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 ... | $$12 \sqrt{2}$$ |
218 | 0.2 | [
{
"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 s... | ①②③④ |
223 | 0.2 | [
{
"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... | ②④ |
225 | 0.2 | [
{
"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 $$A D = \\fra... | ②③④ |
230 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$... | ①②③④ |
231 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the side length of square $$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$$ respecti... | $$\frac{2 \sqrt{13}}{3}$$ |
235 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in the equilateral triangle $$\\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$$, d... | $$3 \sqrt{7}$$ $$\frac{\sqrt{39}}{2}$$ |
237 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in the square $$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$$ as the reflect... | 2 or $$2 \sqrt{3}$$ |
265 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ at poi... | $$\frac{3 \sqrt{3}}{2}$$ or $$\frac{3 \sqrt{7}}{2}$$ |
266 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in square $$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$$ at point G, with foot at point M:"
},
{
"type": "image_path",
"image_path": "image... | 1 $$\sqrt{10} - \sqrt{2}$$/$$- \sqrt{2} + \sqrt{10}$$ |
279 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in square $$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$$ inter... | $$\sqrt{3}$$ |
283 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ at... | 3 |
291 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, O is the intersection point of the diagonals of rectangle $$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$... | $$\frac{9}{8}$$ |
294 | 0.2 | [
{
"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... | $$\frac{4 \sqrt{10}}{5}$$ |
298 | 0.2 | [
{
"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 follow... | ①②③ |
307 | 0.2 | [
{
"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 ... | $$2 \sqrt{21}$$ |
308 | 0.2 | [
{
"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 , ... | $$6 - 2 \sqrt{3}$$/$$- 2 \sqrt{3} + 6$$ $$2 \sqrt{3} - 2$$/$$- 2 + 2 \sqrt{3}$$ |
309 | 0.2 | [
{
"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 ... | $$5 \sqrt{7} + 5 \sqrt{3}$$ |
317 | 0.2 | [
{
"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... | $$\frac{5 \sqrt{3}}{2} - \frac{5 \sqrt{21}}{7}$$ |
318 | 0.2 | [
{
"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 poin... | $$\frac{7}{5}$$ |
330 | 0.2 | [
{
"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 wi... | 2 or $$4 \sqrt{2} - 4$$ |
334 | 0.2 | [
{
"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 $$BI ... | $$\frac{15}{2}$$ |
335 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in quadrilateral $$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$$ is mini... | $$2 \sqrt{19}$$ |
340 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, it is given that quadrilateral $$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$$ divides the area of $$... | $$\frac{75}{41}$$ or $$\frac{75}{17}$$ |
343 | 0.2 | [
{
"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 min... | $$\sqrt{29}$$ |
346 | 0.2 | [
{
"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 $$C F... | 4 or $$\frac{1}{4}$$ |
352 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ is ."
},
{
"type": "image_path",
"image_path... | 5 |
360 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the radius of ⊙O is 1, point B is the midpoint of radius OA, and point C is a moving point on ⊙O. Rotate CB clockwise by 90° around point B to obtain BD. Let M be the midpoint of AD. Then the maximum value of OM is ______."
},
{
"type": "image_path"... | $$\frac{1}{2} + \frac{\sqrt{10}}{4}$$ |
369 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, $$\\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$$, then the area ... | $$4 \sqrt{3}$$; $$6 + 4 \sqrt{3}$$/$$4 \sqrt{3} + 6$$. |
370 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, given $$\\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$$ on the right side... | $$4 \sqrt{3}$$ $$\frac{4}{3} \sqrt{3}$$ |
371 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$... | $$2 \sqrt{21}$$ |
372 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, the diameter $$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$$ is , and the maximum value o... | 8 $$2 \sqrt{11}$$ |
374 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, $$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$$ at poin... | $$2 \sqrt{5}$$ |
376 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in the rhombus $$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}{... | $$\frac{4}{3}$$/$$1 \frac{1}{3}$$ |
381 | 0.2 | [
{
"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 ."
},
{
"ty... | 3 |
388 | 0.2 | [
{
"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... | $$3 \sqrt{5} - 1$$/$$- 1 + 3 \sqrt{5}$$ |
389 | 0.2 | [
{
"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... | $$\frac{\sqrt{13} - \sqrt{5}}{2}$$ |
392 | 0.2 | [
{
"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 ... | $$\frac{225}{8}$$ |
395 | 0.2 | [
{
"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 $$E^{'}$$, $$A D$$ in... | $$\frac{27}{2}$$ |
398 | 0.2 | [
{
"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 le... | $$\frac{2 \sqrt{41} - 5}{2}$$ |
403 | 0.2 | [
{
"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 $$90 ... | 3 or 5/5 or 3 |
408 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, there is a moving point E on side BC of rectangle ABCD. Connect AE, and construct rectangle AEGF with AE as one side, such that side FG passes through point D. If AB = √3 and BC = 4, when triangle AED is an isosceles triangle, the length of BE is ."
... | 2 or $$\sqrt{13}$$ or $$4 - \sqrt{13}$$ |
409 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$. Let $$... | 2 |
413 | 0.2 | [
{
"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 ... | $$\sqrt{10} - \frac{9}{4}$$ |
415 | 0.2 | [
{
"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 $$B F$$,... | $$5 \sqrt{3}$$ |
417 | 0.2 | [
{
"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... | $$\frac{7}{5}$$ or $$\frac{16}{5}$$ |
418 | 0.2 | [
{
"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 $$D P$$, $$C P$$.... | $$8$$ or $$4$$ |
419 | 0.2 | [
{
"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 ... | $$\frac{18 \sqrt{10}}{7}$$/$$\frac{18}{7} \sqrt{10}$$ |
421 | 0.2 | [
{
"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 $$AE = AF =... | $$\frac{40}{7}$$/$$5 \frac{5}{7}$$ |
422 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in rectangle $$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$$ is ... | $$2 \sqrt{13} + 4$$ |
425 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$.... | 4 $$8 + 4 \sqrt{10}$$ |
429 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, it is given that quadrilateral $$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$$, and le... | $$2$$ $$3$$ |
433 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in the rhombus $$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 =$$ ... | $$\frac{3}{2}$$ or $$\frac{6}{5}$$ |
436 | 0.2 | [
{
"type": "text",
"text": "As shown in the figure, in $$\\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$$ is ."
},
{
... | $$45 \circ$$ |
End of preview. Expand in Data Studio
ExamGeo
ExamGeo contains 1,000 English plane-geometry problems for multimodal evaluation. Each record includes a problem statement, one diagram, a reference answer, and a difficulty value.
Contents
data.jsonl 1,000 problems
images/ 1,000 diagrams
Each line in data.jsonl has this format:
{
"id": "3",
"difficulty": "0.2",
"question_list": [
{"type": "text", "text": "..."},
{"type": "image_path", "image_path": "images/3_q0.png"}
],
"answer": "①②③④⑤"
}
Image paths are relative to the ExamGeo directory.
Load
from datasets import load_dataset
dataset = load_dataset("json", data_files="data.jsonl", split="train")
example = dataset[0]
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