fuyyy74/K12_2k1 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 26 915	answer stringlengths 1 100
	<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ABC=70^{\circ}$$, the external angle bisector of $$\angle BAC$$ intersects with the external angle bisector of $$\angle ACB$$ at point $$O$$. Then, $$\angle ABO=$$ ___ degrees.	35
	<image>As shown in the figure, the two points $$A$$ and $$B$$ on the number line represent the numbers $$a$$ and $$b$$, respectively. Among $$a+b$$, $$a-b$$, $$ab$$, and $$\left \lvert a\right \rvert-\left \lvert b\right \rvert$$, the number of positive values is ___.	1
	<image>As shown in the figure, the diagonal $AC$ of quadrilateral $ABCD$ is the perpendicular bisector of $BD$, and $AB=5$, $BC=3$. What is the perimeter of quadrilateral $ABCD$?	16
	<image>As shown in the figure, a circle with a diameter of 1 unit rolls to the right along the number line for one complete revolution. A point on the circle moves from the origin O to point O'. The real number corresponding to point O' is:	\pi
	<image>As shown in the figure, in $$\square ABCD$$, $$\angle C=40^{\circ}$$, a perpendicular line is drawn from point $$D$$ to $$AD$$, intersecting $$AB$$ at point $$E$$ and the extension of $$CB$$ at point $$F$$. The measure of $$\angle BEF$$ is ___.	50
	<image>As shown in the figure, in $\Delta ABC$, $\angle ACB=90{}^\circ $, $AC=3$, $BC=4$, and $CD$ is the median of $\Delta ABC$. What is the area of $\Delta ACD$?	3
	<image>In the rectangle $$ABCD$$, $$M$$ and $$N$$ are the midpoints of sides $$AD$$ and $$BC$$, respectively. $$E$$ and $$F$$ are the midpoints of segments $$BM$$ and $$CM$$, respectively. If $$AB=8$$ and $$AD=12$$, what is the perimeter of quadrilateral $$ENFM$$?	20
	<image>Given the graph of the quadratic function y = ax$^{2}$ + bx + c (a ≠ 0) as shown, the solution to the quadratic inequality ax$^{2}$ + bx + c > 0 is:	-1 < x < 3
	<image>As shown in the figure, two squares with side lengths of $\sqrt{3}$ are cut along their diagonals, and the resulting four triangles are rearranged to form a larger square. What is the side length of this larger square?	\sqrt{6}
	<image>As shown in the figure below, two triangles are drawn in five squares with equal side lengths. If the area of triangle $$A$$ is $$45$$ square centimeters, then the area of triangle $$B$$ is ______ square centimeters.	90
	<image>The flowchart is shown in the figure below, then the output result is ___.	127
	<image>In the figure, in $\Delta ABC$, $AB=AC$, $AD\bot BC$ at $D$, points $E$ and $F$ are the trisection points of $AD$. If the area of $\Delta ABC$ is $14cm^2$, then the area of the shaded part in the figure is $cm^2$.	7
	<image>Execute the flowchart shown in the figure. If the input is $$t \in [-1,3]$$, then the range of the output $$s$$ is ___.	[-3,4]
	<image>As shown in the figure, the graphs of the functions $y=-3x$ and $y=ax+4$ intersect at point A (m, 3). The solution set of the inequality $-3x > ax+4$ is.	x < -1
	<image>Rectangle $ABCD$ is folded along $AE$, such that point $D$ lands on point $D'$, resulting in the figure shown. Given that $\angle CE{D}'=60{}^\circ$, what is the measure of $\angle AED$?	{{60}^{{}^\circ }}
	<image>In the cube shown in the figure, $$M$$ and $$N$$ are the midpoints of edges $$BC$$ and $$CC_{1}$$, respectively. The angle formed by the skew lines $$AC$$ and $$MN$$ is ___.	60^{\circ}
	<image>As shown in the figure, point C is on line segment AB, AB - BC = 10 cm, BC:AC = 3:5, then the length of BC is ______ cm.	6
	<image>Read and understand the method of factoring the quadratic expression $$2x^2-x-3$$ using the 'Cross Multiplication Method'. (1) The coefficient of the quadratic term $$2=1×2$$. (2) The constant term $$-3=-1×3=1×(-3)$$, verify the 'sum of cross products'. $$1×3+2×(-1)=1$$, $$1×(-1)+2×3=5$$, $$1×(-3)+2×1=-1$$, $$1×1+2×(-3)=-5$$. (3) It is found that the result of the third 'sum of cross products' $$1×(-3)+2×1=-1$$ equals the coefficient of the linear term $$-1$$. That is, $$(x+1)(2x-3)=2x^2-3x+2x-3=2x^2-x-3$$, so $$2x^2-x-3=(x+1)(2x-3)$$. In this way, the method of factoring a quadratic trinomial using the cross multiplication method is called the 'Cross Multiplication Method'. Following the above method, factor the expression: $$3x^2+5x-12=$$______.	(x+3)(3x-4)
	<image>As shown in the figure, the volume of a regular quadrilateral pyramid with all edges of length 2 is.	\frac{4\sqrt{2}}{3}
	<image>As shown in the figure, in parallelogram $$ABCD$$, point $$O$$ is the midpoint of $$AC$$, and point $$N$$ is the midpoint of $$OB$$. Let $$\overrightarrow{AB} = \boldsymbol{a}$$ and $$\overrightarrow{AD} = \boldsymbol{b}$$. If $$\overrightarrow{AN}$$ is expressed in terms of $$\boldsymbol{a}$$ and $$\boldsymbol{b}$$, then $$\overrightarrow{AN} = $$___.	\dfrac{3}{4}\boldsymbol{a}+\dfrac{1}{4}\boldsymbol{b}
	<image>As shown in the figure, the sides AB, BC, CD, and DA of quadrilateral ABCD are tangent to circle O at points L, M, N, and P, respectively, and AB = 10 cm, CD = 5 cm. What is the perimeter of quadrilateral ABCD in cm?	30
	<image>As shown in the figure, quadrilateral $$ABCD$$ is a rhombus, and $$O$$ is the intersection point of the two diagonals. Three lines passing through point $$O$$ divide the rhombus into shaded and unshaded parts. If the lengths of the two diagonals of the rhombus are $$6$$ and $$8$$, then the area of the shaded part is ___.	12
	<image>As shown in the figure, the diagonals $AC$ and $BD$ of rectangle $ABCD$ intersect at point $O$. A line passing through point $O$ intersects $AD$ and $BC$ at points $E$ and $F$, respectively. Given $AB=3$ and $AC=5$, what is the area of the shaded region?	6
	<image>As shown in the figure, the central angle $$\angle AOB=20^{\circ}$$, and the arc $$\overset{\frown} {AB}$$ is rotated by $$n^{\circ}$$ to get the arc $$\overset{\frown} {CD}$$, then the degree measure of the arc $$\overset{\frown} {CD}$$ is ___ degrees.	20
	<image>In the figure, $PA$ and $PB$ are tangents to circle $\odot O$, with $A$ and $B$ being the points of tangency. Connect $OA$, $AB$, and $\angle OAB=38{}^\circ$. Then $\angle P=$ degrees.	76
	<image>As shown in the figure, there are two slides of the same length (i.e., BC=EF). The height AC of the left slide is equal to the horizontal length DF of the right slide. Then, ∠ABC + ∠DFE = ___ degrees	90
	<image>As shown in the figure, ∠B = ∠C, AC intersects BD at point O. If AO = 2, DO = 1.5, and AB = 4, then the length of CD is.	3
	<image>To estimate the area of the shaded region as shown in the figure, a square with a side length of $$6$$ is drawn to contain it, and $$800$$ points are randomly thrown into the square. It is known that exactly $$200$$ points fall within the shaded region. According to this, the estimated area of the shaded region is ___.	9
	<image>As shown in the figure, the right triangle ABC is rotated 40° counterclockwise around point A to get the right triangle AB′C′, with point C′ exactly landing on side AB. Connecting BB′, what is the measure of ∠BB′C′ in degrees?	20
	<image>As shown in the figure, in $$\triangle ABC$$, $$AB=AC$$, $$AD \perp BC$$, with the foot of the perpendicular at $$D$$, and $$E$$ is the midpoint of $$AC$$. If $$DE=2$$, then the length of $$AB$$ is ___.	4
	<image>As shown in the figure, observe the pattern formed by small cubes with edge lengths of $$1$$. In Figure 1, there is a total of $$1$$ small cube, of which $$1$$ is visible and $$0$$ are invisible; in Figure 2, there are a total of $$8$$ small cubes, of which $$7$$ are visible and $$1$$ is invisible; in Figure 3, there are a total of $$27$$ small cubes, of which $$19$$ are visible and $$8$$ are invisible $$\cdots\cdots$$ Therefore, in Figure 6, the number of visible small cubes is ___ .	91
	<image>As shown in the figure, $$AB \bot CD$$ at point $$B$$, $$BE$$ is the bisector of $$∠ABD$$. The measure of $$∠CBE$$ is ___.	135^{\circ}
	<image>As shown in the figure, at 6:30 AM, the acute angle between the hour hand and the minute hand is.	15°
	<image>As shown in the figure, from a point P outside circle O, two tangents PA and PB are drawn to circle O, touching at points A and B, respectively. If PA = 8 cm, and C is a moving point on arc $\overset\frown{AB}$ (point C does not coincide with points A or B), a tangent is drawn from point C to circle O, intersecting PA and PB at points D and E, respectively. Then the perimeter of triangle PED is cm.	16
	<image>In the figure, in △ABC, AB=AC, ∠BAC=40°, AD is the median, BE is the altitude, AD intersects BE at point F, then ∠AFE=．	70°
	<image>Represent the irrational numbers $\sqrt{11}$, $\sqrt{5}$, and $-\sqrt{3}$ on a number line. Among these three irrational numbers, the one covered by the ink (as shown in the figure) is.	\sqrt{11}
	<image>In the figure, in △ABC, ∠C = 90°, AD bisects ∠BAC, BC = 10 cm, BD = 6 cm, then the distance from point D to AB is ______ cm.	4
	<image>As shown in the figure, OA=3OB, then the length of $\overset\frown{AD}$ is how many times the length of $\overset\frown{BC}$.	3
	<image>In a planar quadrilateral ABCD, AB=AD=CD=1, BD=√2, and BD⊥CD. When folded along the diagonal BD to form a tetrahedron A′-BCD, the plane A′BD is perpendicular to the plane BCD. If the vertices of the tetrahedron A′-BCD lie on the same sphere, find the surface area of the sphere.	3π
	<image>As shown in the figure, the bisector BF of ∠ABC intersects the bisector CF of the exterior angle ∠ACG of △ABC at point F. A line DF parallel to BC is drawn through F, intersecting AB at D and AC at E. If BD = 8 and DE = 3, then the length of CE is;	5
	<image>Given the probability distribution of a discrete random variable $$X$$, the variance of the random variable $$X$$, $$V(X)$$, is equal to ___.	\dfrac{2}{9}
	<image>As shown in the figure, the number of triangles in the figure is ___.	9
	<image>As shown in the figure, in the Cartesian coordinate system $xOy$, an ellipse and a hyperbola centered at the origin intersect at four points $A, B, C, D$, and they have the same foci $F_1, F_2$. Points $F_1, F_2$ lie on $AD$ and $BC$ respectively. Then the product of the eccentricities of the ellipse and the hyperbola $e_1 \cdot e_2 =$	1
	<image>As shown in the figure, the radius OA of circle O is 5, the radius OC is perpendicular to the chord AB, and the foot of the perpendicular is D. If OD = 3, then the length of chord AB is.	8
	<image>As shown in the figure, a ruler is placed on a triangular board (ignoring thickness). One side of the ruler, $$MN$$, intersects the two sides of the triangle, $$AB$$ and $$AC$$, at points $$E$$ and $$F$$ respectively. Given that $$\angle A=30^{ \circ }$$, what is the degree measure of $$\angle AEM+ \angle AFN$$?	210^{ \circ }
	<image>As shown in the figure, given that $\text{EF} \parallel \text{BC}$, $\text{AE}=3$, $\text{BE}=4$, and $\text{FC}=6$, then the value of $\text{AF}$ is.	\frac{9}{2}
	<image>As shown in the figure, quadrilateral $$ABCD$$ is inscribed in circle $$ \odot O$$, $$ \angle BCD=100^{ \circ }$$, and $$AC$$ bisects $$ \angle BAD$$. What is the measure of $$ \angle BAC$$?	40^{\circ}
	<image>As shown in the figure, the radius of $$\odot O$$ is $$6$$, and the angle between $$OA$$ and the chord $$AB$$ is $$30^{\circ}$$, then the length of the chord $$AB$$ is ___.	6\sqrt{3}
	<image>In the figure, in the right triangle $\text{Rt}\Delta ABC$, point $D$ is the midpoint of $AB$, $CD=5$, and $BC=8$. Then, $AC=$.	6
	<image>The parabola shown in the figure is the graph of the quadratic function $$y=ax^2-3x+a^2-1$$. What is the value of $$a$$?	-1
	<image>In the right triangle ABC, ∠ACB = 90°, the area of the square with side AC is 15, and the length of the median CD is 2. What is the length of BC?	1
	<image>If the tetrahedron $$P-ABC$$ (as shown in figure (1)) is cut along $$PA$$, $$PB$$, and $$PC$$, a lateral development diagram (as shown in figure (2)) is obtained in the plane of $$\triangle ABC$$. In this diagram, $$P_{1}$$, $$B$$, and $$P_{2}$$ are collinear, $$P_{3}$$, $$C$$, and $$P_{2}$$ are collinear, and $$P_{2}P_{1}=P_{2}P_{3}$$. What is the measure of the angle formed by the skew lines $$PA$$ and $$BC$$ in $$P-ABC$$?	90^{ \circ }
	<image>The statistical data for the advertising expenses $$x$$ (in ten thousand yuan) and sales $$y$$ (in ten thousand yuan) of a product are shown in the following table: According to the table, the regression equation $$\hat{y}=\hat{b}x+\hat{a}$$ has $$\widehat{b}$$ as $$7$$. Based on this model, predict the sales when the advertising expenses are $$10$$ ten thousand yuan.	73.5
	<image>As shown in the figure, AC is the diagonal of square ABCD with side length 1, point E is a point on the ray CB, and CE = CA, then EB =.	\sqrt{2}-1
	<image>As shown in the figure, in rectangle ABCD, ∠DAC=65°. Point E is a point on CD, and BE intersects AC at point F. When △BCE is folded along BE, point C exactly lands on point C′ on AB. Find ∠AFC′.	40{}^\circ
	<image>As shown in the figure, plane $$ \alpha \bot $$ plane $$ \beta $$, $$A \in \alpha $$, $$B \in \beta $$, $$AA'\bot A' B'$$, $$BB'\bot A'B'$$, and $$AA'= 3$$, $$BB'= 4$$, $$A'B'= 2$$. What is the volume $$V$$ of the tetrahedron $$A - A'BB'$$?	4
	<image>As shown in the figure, observe the following group of shapes. In the first shape, there are 2 stars; in the second shape, there are 6 stars; in the third shape, there are 11 stars; in the fourth shape, there are 17 stars... Following this pattern, the number of stars in the eighth shape is .	51
	<image>The result output by the following program is ___.	0
	<image>As shown in the figure, write a number at each vertex of the square $$ABCD$$. Add the numbers at the two endpoints of each side of the square, and write the sum on that side. It is known that the number on side $$AB$$ is $$3$$, the number on side $$BC$$ is $$7$$, and the number on side $$CD$$ is $$12$$. What is the number on side $$AD$$?	8
	<image>As shown in the figure, in a cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$ with edge length $$2$$, $$E$$ and $$F$$ are the midpoints of edges $$AB$$ and $$BC$$, respectively. The volume of the tetrahedron $$B-B_{1}EF$$ is ___.	\dfrac{1}{3}
	<image>As shown in the figure, a square $$BC-DE$$ is constructed with one unit length from the number line as its side. A circle is drawn with the point representing $$1$$ on the number line as the center and the length of the diagonal of the square as the radius, intersecting the negative half of the number line at point $$A$$. The number represented by point $$A$$ is ___.	1-\sqrt{2}
	<image>As shown in the figure, the area of rectangle OABC is 18. Its diagonal OB intersects the hyperbola y=$\frac{k}{x}$ at point D, and OD:DB=2:1, then k=．	8
	<image>As shown in the figure, O is a point on the straight line l, ∠1 + ∠2 = 78°42′, then ∠AOB =.	101°18′
	<image>As shown in the figure, there is a square plastic template ABCD with a side length of 4. The right-angle vertex of a sufficiently large right-angled triangle is placed at point A, with its two legs intersecting CD at point F and the extension of CB at point E. The area of quadrilateral AECF is ______.	16
	<image>As shown in the figure, water is transferred from bucket A to bucket B. Initially, bucket A has a liters of water. After t minutes, the remaining water y liters satisfies the function y=ae^{-nt}. Therefore, the water in bucket B is y=a-ae^{-nt}. Assuming that after 5 minutes, the water in bucket A and bucket B is equal, then after _____ minutes, the water in bucket A will be \frac{a}{8}L.	10
	<image>As shown in the figure, triangle ABC is translated 3cm to the left to obtain triangle DEF, where points E, B, F, and C lie on the same straight line. If the perimeter of triangle ABC is 12cm, then the perimeter of quadrilateral ACED is cm.	18cm
	<image>Fold the right figure into a cube, the maximum sum of the numbers on the opposite faces is ______.	9
	<image>As shown in the figure, given $$AB\parallel DE$$, $$BC\parallel EF$$, then the number of pairs of homothetic triangles in the figure is ___.	4
	<image>As shown in the figure, in $$\triangle ABC$$, point $$O$$ is the midpoint of $$BC$$. A line through point $$O$$ intersects the lines $$AB$$ and $$AC$$ at two distinct points $$M$$ and $$N$$, respectively. If $$\overrightarrow{AB}=m\overrightarrow{AM}$$ and $$\overrightarrow{AC}=n\overrightarrow{AN}$$, then the value of $$m+n$$ is ___.	2
	<image>As shown in the figure, there is some water in a cubic container with an edge length of 10 cm, and the water height is 7 cm. A rectangular iron block, 8 cm in height, is placed vertically into the water (the base of the iron block is parallel to the bottom of the container). Before the iron block is completely submerged, the water is already full, and the submerged part of the iron block is 6 cm high. The volume of this iron block is ______ cubic centimeters.	400
	<image>In the flowchart shown, the output value of $n$ is.	4
	<image>As shown in the figure, the perimeter of ▱ABCD is 18 cm, and the diagonals AC and BD intersect at point O. If the difference in the perimeters of △AOD and △AOB is 5 cm, then the length of side AB is ______ cm.	2
	<image>Place a small ball at the top entrance of the container shown in the figure, and the ball will fall freely. During its fall, the ball will encounter a black obstacle 3 times, and it will finally fall into bag A or bag B. It is known that each time the ball encounters a black obstacle, the probability of falling to the left or right is $$\dfrac{1}{2}$$, then the probability of the ball falling into bag A is ___.	\dfrac{3}{4}
	<image>In the figure, in △ABC, AB = 8, AC = 6, AD and AE are the angle bisector and median respectively. A perpendicular CG is drawn from point C to AD at F, intersecting AB at G. EF is connected. Find the length of EF.	1
	<image>Given y = , then the arithmetic square root of xy is ______.	6
	<image>In triangle ABC, ∠ABC = 2∠C, AP and BQ are the angle bisectors of ∠BAC and ∠ABC, respectively. If the perimeter of triangle ABQ is 18, and BP = 4, then the length of AB is	7
	<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ACB=90^{\circ}$$, point $$F$$ is on the extension of $$AC$$ such that $$CF=\dfrac {1}{2}AC$$, and $$DE$$ is the midsegment of $$\triangle ABC$$. If $$\angle 1=30^{\circ}$$ and $$DE=2$$, then the perimeter of quadrilateral $$AFED$$ is ___.	16
	<image>As shown in the figure, the result output by the algorithm is ___.	21
	<image>As shown in the figure, $$AB$$ is the diameter of $$\odot O$$, and chord $$CD$$ perpendicularly bisects $$OB$$ at point $$E$$. Connecting $$OD$$ and $$BC$$, if $$BC=1$$, then the area of sector $$OBD$$ is ___.	\dfrac{\pi }{6}
	<image>As shown in the figure, the area of △ABC is 24, D is the midpoint of BC, and E is the midpoint of AC. What is the area of △CDE?	6
	<image>If the following program flowchart is executed, then the output $S=$.	20
	<image>As shown in the figure, in circle O, AB is the diameter, and point C is on circle O. The bisector of ∠ACB intersects circle O at D. Then ∠ABD = {{ }}°.	45
	<image>The program in the figure calculates the function value. If the input value of x is $\frac{3}{2}$, then the output result y is.	0.5
	<image>As shown in the figure, place right triangle ABC in a rectangular coordinate system, where ∠CAB = 90°, BC = 5, and the coordinates of points A and B are (﹣1, 0) and (﹣4, 0), respectively. Triangle ABC is translated to the left along the x-axis. When point C falls on the line y = -2x - 6, how many units has point C been translated to the left along the x-axis?	4
	<image>As shown in the figure, line l is perpendicular to the x-axis at point P, and intersects the graphs of the inverse proportion functions y_1 = k_1 / x (x > 0) and y_2 = k_2 / x (x > 0) at points A and B, respectively. Connecting OA and OB, the area of triangle OAB is known to be 1. Then k_1 - k_2 =.	2
	<image>The following is a statistical graph of Xiao Hua's scores in five math tests. The average score of Xiao Hua's five tests is ______ points.	92
	<image>Among the following four car logos, the number of logos that are both centrally symmetric and axially symmetric is ______.	1
	<image>During the flag-raising ceremony at the Chinese National Day parade, as shown in the figure, a row of seats and the flagpole are on the same vertical plane perpendicular to the ground on a viewing stand with an incline of 15°. The angles of elevation to the top of the flagpole from the first and last rows are 60° and 30°, respectively, and the distance between the first and last rows is 10\sqrt{6}m. What is the height of the flagpole?	30
	<image>As shown in the figure, let $$G$$ be the centroid of $$ \triangle ABC$$, and let the midpoints of $$BC$$, $$CA$$, and $$AB$$ be $$D$$, $$E$$, and $$F$$, respectively. Then $$\overrightarrow{GD} + \overrightarrow{GE} + \overrightarrow{GF} = $$___.	\overrightarrow{0}
	<image>In triangle ABC, it is known that AB = AC, BC = BD, AD = DE = EB, the measure of angle A is ______.	45°
	<image>Given the power function $$f(x)=x^{a}$$ with some corresponding values as shown in the table below, then the solution set of the inequality $$f(\|x\|) \leqslant 2$$ is ___.	\{x\|-4 \leqslant x \leqslant 4\}
	<image>Determine whether the input number $$x$$ is positive. If it is, output its square; if not, output its opposite. The blank should be filled with ___.	x \leqslant 0
	<image>In the figure, in $$\triangle ABC$$, $$AC=BC=10cm$$, $$\angle A=30\degree$$, $$CD$$ is the median of $$\triangle ABC$$, and a line through $$D$$ parallel to $$BC$$ intersects the bisector of $$\angle BCD$$ at point $$E$$. The length of $$DE$$ is ______.	5
	<image>In the right triangle ABC, AC⊥BC, and CD⊥AB at D, AD=4, BD=2, then CD=．	2\sqrt{2}
	<image>As shown in the figure, points $$P$$ and $$Q$$ are on the graph of the inverse proportion function $$y=\dfrac{k}{x}$$. $$PA \perp y$$-axis at point $$A$$, $$QN \perp x$$-axis at point $$N$$, $$PM \perp x$$-axis at point $$M$$, and $$QB \perp y$$-axis at point $$B$$. Connecting $$PB$$ and $$QM$$, if the area of $$\triangle ABP$$ is denoted as $$S_{1}$$, and the area of $$\triangle QMN$$ is denoted as $$S_{2}$$, then $$S_{1}$$ ___ $$S_{2}$$ (fill in "$$>$$", "$$<$$", or "$$=$$").	=
	<image>As shown in the figure, given that the edge length of the cube $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ is a, the angle formed by the skew lines $B{{C}_{1}}$ and $AC$ is．	60{}^\circ
	<image>As shown in the figure, the lines $$AD∥BE$$, $$AC$$ and $$BC$$ bisect $$∠BAD$$ and $$∠ABE$$ respectively, and $$∠CAD=55^{\circ}$$. Then $$∠CBE=$$ ___.	35^{\circ}
	<image>As shown in the figure, a soybean is randomly scattered in a square with a side length of 1. The probability that it lands in the shaded area is.	\frac{1}{3}
	<image>With the continuous advancement of spiritual civilization, citizens are spending more and more time each day on reading books, reading newspapers, and participating in 'National Fitness Campaigns.' The figure below is a frequency distribution histogram created by a reporter from our city's evening newspaper after a sample survey of some citizens' time spent on these activities. The sum of the areas of the first seven rectangles from left to right is $$\number{0.94}$$, and the frequency of the last group is $$12$$. What is the sample size of this survey?	200
	<image>As shown in the figure, $OP$ bisects $\angle AOB$, $PE \perp AO$ at point $E$, $PF \perp BO$ at point $F$, and $PE = 6\text{cm}$. Then the distance from point $P$ to $OB$ is $\text{cm}$.	6

<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ABC=70^{\circ}$$, the external angle bisector of $$\angle BAC$$ intersects with the external angle bisector of $$\angle ACB$$ at point $$O$$. Then, $$\angle ABO=$$ ___ degrees.

35

<image>As shown in the figure, the two points $$A$$ and $$B$$ on the number line represent the numbers $$a$$ and $$b$$, respectively. Among $$a+b$$, $$a-b$$, $$ab$$, and $$\left \lvert a\right \rvert-\left \lvert b\right \rvert$$, the number of positive values is ___.

1

<image>As shown in the figure, the diagonal $AC$ of quadrilateral $ABCD$ is the perpendicular bisector of $BD$, and $AB=5$, $BC=3$. What is the perimeter of quadrilateral $ABCD$?

16

<image>As shown in the figure, a circle with a diameter of 1 unit rolls to the right along the number line for one complete revolution. A point on the circle moves from the origin O to point O'. The real number corresponding to point O' is:

\pi

<image>As shown in the figure, in $$\square ABCD$$, $$\angle C=40^{\circ}$$, a perpendicular line is drawn from point $$D$$ to $$AD$$, intersecting $$AB$$ at point $$E$$ and the extension of $$CB$$ at point $$F$$. The measure of $$\angle BEF$$ is ___.

50

<image>As shown in the figure, in $\Delta ABC$, $\angle ACB=90{}^\circ $, $AC=3$, $BC=4$, and $CD$ is the median of $\Delta ABC$. What is the area of $\Delta ACD$?

3

<image>In the rectangle $$ABCD$$, $$M$$ and $$N$$ are the midpoints of sides $$AD$$ and $$BC$$, respectively. $$E$$ and $$F$$ are the midpoints of segments $$BM$$ and $$CM$$, respectively. If $$AB=8$$ and $$AD=12$$, what is the perimeter of quadrilateral $$ENFM$$?

20

<image>Given the graph of the quadratic function y = ax$^{2}$ + bx + c (a ≠ 0) as shown, the solution to the quadratic inequality ax$^{2}$ + bx + c > 0 is:

-1 < x < 3

<image>As shown in the figure, two squares with side lengths of $\sqrt{3}$ are cut along their diagonals, and the resulting four triangles are rearranged to form a larger square. What is the side length of this larger square?

\sqrt{6}

<image>As shown in the figure below, two triangles are drawn in five squares with equal side lengths. If the area of triangle $$A$$ is $$45$$ square centimeters, then the area of triangle $$B$$ is ______ square centimeters.

90

<image>The flowchart is shown in the figure below, then the output result is ___.

127

<image>In the figure, in $\Delta ABC$, $AB=AC$, $AD\bot BC$ at $D$, points $E$ and $F$ are the trisection points of $AD$. If the area of $\Delta ABC$ is $14cm^2$, then the area of the shaded part in the figure is $cm^2$.

7

<image>Execute the flowchart shown in the figure. If the input is $$t \in [-1,3]$$, then the range of the output $$s$$ is ___.

[-3,4]

<image>As shown in the figure, the graphs of the functions $y=-3x$ and $y=ax+4$ intersect at point A (m, 3). The solution set of the inequality $-3x > ax+4$ is.

x < -1

<image>Rectangle $ABCD$ is folded along $AE$, such that point $D$ lands on point $D'$, resulting in the figure shown. Given that $\angle CE{D}'=60{}^\circ$, what is the measure of $\angle AED$?

{{60}^{{}^\circ }}

<image>In the cube shown in the figure, $$M$$ and $$N$$ are the midpoints of edges $$BC$$ and $$CC_{1}$$, respectively. The angle formed by the skew lines $$AC$$ and $$MN$$ is ___.

60^{\circ}

<image>As shown in the figure, point C is on line segment AB, AB - BC = 10 cm, BC:AC = 3:5, then the length of BC is ______ cm.

6

<image>Read and understand the method of factoring the quadratic expression $$2x^2-x-3$$ using the 'Cross Multiplication Method'. (1) The coefficient of the quadratic term $$2=1×2$$. (2) The constant term $$-3=-1×3=1×(-3)$$, verify the 'sum of cross products'. $$1×3+2×(-1)=1$$, $$1×(-1)+2×3=5$$, $$1×(-3)+2×1=-1$$, $$1×1+2×(-3)=-5$$. (3) It is found that the result of the third 'sum of cross products' $$1×(-3)+2×1=-1$$ equals the coefficient of the linear term $$-1$$. That is, $$(x+1)(2x-3)=2x^2-3x+2x-3=2x^2-x-3$$, so $$2x^2-x-3=(x+1)(2x-3)$$. In this way, the method of factoring a quadratic trinomial using the cross multiplication method is called the 'Cross Multiplication Method'. Following the above method, factor the expression: $$3x^2+5x-12=$$______.

(x+3)(3x-4)

<image>As shown in the figure, the volume of a regular quadrilateral pyramid with all edges of length 2 is.

\frac{4\sqrt{2}}{3}

<image>As shown in the figure, in parallelogram $$ABCD$$, point $$O$$ is the midpoint of $$AC$$, and point $$N$$ is the midpoint of $$OB$$. Let $$\overrightarrow{AB} = \boldsymbol{a}$$ and $$\overrightarrow{AD} = \boldsymbol{b}$$. If $$\overrightarrow{AN}$$ is expressed in terms of $$\boldsymbol{a}$$ and $$\boldsymbol{b}$$, then $$\overrightarrow{AN} = $$___.

\dfrac{3}{4}\boldsymbol{a}+\dfrac{1}{4}\boldsymbol{b}

<image>As shown in the figure, the sides AB, BC, CD, and DA of quadrilateral ABCD are tangent to circle O at points L, M, N, and P, respectively, and AB = 10 cm, CD = 5 cm. What is the perimeter of quadrilateral ABCD in cm?

30

<image>As shown in the figure, quadrilateral $$ABCD$$ is a rhombus, and $$O$$ is the intersection point of the two diagonals. Three lines passing through point $$O$$ divide the rhombus into shaded and unshaded parts. If the lengths of the two diagonals of the rhombus are $$6$$ and $$8$$, then the area of the shaded part is ___.

12

<image>As shown in the figure, the diagonals $AC$ and $BD$ of rectangle $ABCD$ intersect at point $O$. A line passing through point $O$ intersects $AD$ and $BC$ at points $E$ and $F$, respectively. Given $AB=3$ and $AC=5$, what is the area of the shaded region?

6

<image>As shown in the figure, the central angle $$\angle AOB=20^{\circ}$$, and the arc $$\overset{\frown} {AB}$$ is rotated by $$n^{\circ}$$ to get the arc $$\overset{\frown} {CD}$$, then the degree measure of the arc $$\overset{\frown} {CD}$$ is ___ degrees.

20

<image>In the figure, $PA$ and $PB$ are tangents to circle $\odot O$, with $A$ and $B$ being the points of tangency. Connect $OA$, $AB$, and $\angle OAB=38{}^\circ$. Then $\angle P=$ degrees.

76

<image>As shown in the figure, there are two slides of the same length (i.e., BC=EF). The height AC of the left slide is equal to the horizontal length DF of the right slide. Then, ∠ABC + ∠DFE = ___ degrees

90

<image>As shown in the figure, ∠B = ∠C, AC intersects BD at point O. If AO = 2, DO = 1.5, and AB = 4, then the length of CD is.

3

<image>To estimate the area of the shaded region as shown in the figure, a square with a side length of $$6$$ is drawn to contain it, and $$800$$ points are randomly thrown into the square. It is known that exactly $$200$$ points fall within the shaded region. According to this, the estimated area of the shaded region is ___.

9

<image>As shown in the figure, the right triangle ABC is rotated 40° counterclockwise around point A to get the right triangle AB′C′, with point C′ exactly landing on side AB. Connecting BB′, what is the measure of ∠BB′C′ in degrees?

20

<image>As shown in the figure, in $$\triangle ABC$$, $$AB=AC$$, $$AD \perp BC$$, with the foot of the perpendicular at $$D$$, and $$E$$ is the midpoint of $$AC$$. If $$DE=2$$, then the length of $$AB$$ is ___.

4

<image>As shown in the figure, observe the pattern formed by small cubes with edge lengths of $$1$$. In Figure 1, there is a total of $$1$$ small cube, of which $$1$$ is visible and $$0$$ are invisible; in Figure 2, there are a total of $$8$$ small cubes, of which $$7$$ are visible and $$1$$ is invisible; in Figure 3, there are a total of $$27$$ small cubes, of which $$19$$ are visible and $$8$$ are invisible $$\cdots\cdots$$ Therefore, in Figure 6, the number of visible small cubes is ___ .

91

<image>As shown in the figure, $$AB \bot CD$$ at point $$B$$, $$BE$$ is the bisector of $$∠ABD$$. The measure of $$∠CBE$$ is ___.

135^{\circ}

<image>As shown in the figure, at 6:30 AM, the acute angle between the hour hand and the minute hand is.

15°

<image>As shown in the figure, from a point P outside circle O, two tangents PA and PB are drawn to circle O, touching at points A and B, respectively. If PA = 8 cm, and C is a moving point on arc $\overset\frown{AB}$ (point C does not coincide with points A or B), a tangent is drawn from point C to circle O, intersecting PA and PB at points D and E, respectively. Then the perimeter of triangle PED is cm.

16

<image>In the figure, in △ABC, AB=AC, ∠BAC=40°, AD is the median, BE is the altitude, AD intersects BE at point F, then ∠AFE=．

70°

<image>Represent the irrational numbers $\sqrt{11}$, $\sqrt{5}$, and $-\sqrt{3}$ on a number line. Among these three irrational numbers, the one covered by the ink (as shown in the figure) is.

\sqrt{11}

<image>In the figure, in △ABC, ∠C = 90°, AD bisects ∠BAC, BC = 10 cm, BD = 6 cm, then the distance from point D to AB is ______ cm.

4

<image>As shown in the figure, OA=3OB, then the length of $\overset\frown{AD}$ is how many times the length of $\overset\frown{BC}$.

3

<image>In a planar quadrilateral ABCD, AB=AD=CD=1, BD=√2, and BD⊥CD. When folded along the diagonal BD to form a tetrahedron A′-BCD, the plane A′BD is perpendicular to the plane BCD. If the vertices of the tetrahedron A′-BCD lie on the same sphere, find the surface area of the sphere.

3π

<image>As shown in the figure, the bisector BF of ∠ABC intersects the bisector CF of the exterior angle ∠ACG of △ABC at point F. A line DF parallel to BC is drawn through F, intersecting AB at D and AC at E. If BD = 8 and DE = 3, then the length of CE is;

5

<image>Given the probability distribution of a discrete random variable $$X$$, the variance of the random variable $$X$$, $$V(X)$$, is equal to ___.

\dfrac{2}{9}

<image>As shown in the figure, the number of triangles in the figure is ___.

9

<image>As shown in the figure, in the Cartesian coordinate system $xOy$, an ellipse and a hyperbola centered at the origin intersect at four points $A, B, C, D$, and they have the same foci $F_1, F_2$. Points $F_1, F_2$ lie on $AD$ and $BC$ respectively. Then the product of the eccentricities of the ellipse and the hyperbola $e_1 \cdot e_2 =$

1

<image>As shown in the figure, the radius OA of circle O is 5, the radius OC is perpendicular to the chord AB, and the foot of the perpendicular is D. If OD = 3, then the length of chord AB is.

8

<image>As shown in the figure, a ruler is placed on a triangular board (ignoring thickness). One side of the ruler, $$MN$$, intersects the two sides of the triangle, $$AB$$ and $$AC$$, at points $$E$$ and $$F$$ respectively. Given that $$\angle A=30^{ \circ }$$, what is the degree measure of $$\angle AEM+ \angle AFN$$?

210^{ \circ }

<image>As shown in the figure, given that $\text{EF} \parallel \text{BC}$, $\text{AE}=3$, $\text{BE}=4$, and $\text{FC}=6$, then the value of $\text{AF}$ is.

\frac{9}{2}

<image>As shown in the figure, quadrilateral $$ABCD$$ is inscribed in circle $$ \odot O$$, $$ \angle BCD=100^{ \circ }$$, and $$AC$$ bisects $$ \angle BAD$$. What is the measure of $$ \angle BAC$$?

40^{\circ}

<image>As shown in the figure, the radius of $$\odot O$$ is $$6$$, and the angle between $$OA$$ and the chord $$AB$$ is $$30^{\circ}$$, then the length of the chord $$AB$$ is ___.

6\sqrt{3}

<image>In the figure, in the right triangle $\text{Rt}\Delta ABC$, point $D$ is the midpoint of $AB$, $CD=5$, and $BC=8$. Then, $AC=$.

6

<image>The parabola shown in the figure is the graph of the quadratic function $$y=ax^2-3x+a^2-1$$. What is the value of $$a$$?

-1

<image>In the right triangle ABC, ∠ACB = 90°, the area of the square with side AC is 15, and the length of the median CD is 2. What is the length of BC?

1

<image>If the tetrahedron $$P-ABC$$ (as shown in figure (1)) is cut along $$PA$$, $$PB$$, and $$PC$$, a lateral development diagram (as shown in figure (2)) is obtained in the plane of $$\triangle ABC$$. In this diagram, $$P_{1}$$, $$B$$, and $$P_{2}$$ are collinear, $$P_{3}$$, $$C$$, and $$P_{2}$$ are collinear, and $$P_{2}P_{1}=P_{2}P_{3}$$. What is the measure of the angle formed by the skew lines $$PA$$ and $$BC$$ in $$P-ABC$$?

90^{ \circ }

<image>The statistical data for the advertising expenses $$x$$ (in ten thousand yuan) and sales $$y$$ (in ten thousand yuan) of a product are shown in the following table: According to the table, the regression equation $$\hat{y}=\hat{b}x+\hat{a}$$ has $$\widehat{b}$$ as $$7$$. Based on this model, predict the sales when the advertising expenses are $$10$$ ten thousand yuan.

73.5

<image>As shown in the figure, AC is the diagonal of square ABCD with side length 1, point E is a point on the ray CB, and CE = CA, then EB =.

\sqrt{2}-1

<image>As shown in the figure, in rectangle ABCD, ∠DAC=65°. Point E is a point on CD, and BE intersects AC at point F. When △BCE is folded along BE, point C exactly lands on point C′ on AB. Find ∠AFC′.

40{}^\circ

<image>As shown in the figure, plane $$ \alpha \bot $$ plane $$ \beta $$, $$A \in \alpha $$, $$B \in \beta $$, $$AA'\bot A' B'$$, $$BB'\bot A'B'$$, and $$AA'= 3$$, $$BB'= 4$$, $$A'B'= 2$$. What is the volume $$V$$ of the tetrahedron $$A - A'BB'$$?

4

<image>As shown in the figure, observe the following group of shapes. In the first shape, there are 2 stars; in the second shape, there are 6 stars; in the third shape, there are 11 stars; in the fourth shape, there are 17 stars... Following this pattern, the number of stars in the eighth shape is .

51

<image>The result output by the following program is ___.

0

<image>As shown in the figure, write a number at each vertex of the square $$ABCD$$. Add the numbers at the two endpoints of each side of the square, and write the sum on that side. It is known that the number on side $$AB$$ is $$3$$, the number on side $$BC$$ is $$7$$, and the number on side $$CD$$ is $$12$$. What is the number on side $$AD$$?

8

<image>As shown in the figure, in a cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$ with edge length $$2$$, $$E$$ and $$F$$ are the midpoints of edges $$AB$$ and $$BC$$, respectively. The volume of the tetrahedron $$B-B_{1}EF$$ is ___.

\dfrac{1}{3}

<image>As shown in the figure, a square $$BC-DE$$ is constructed with one unit length from the number line as its side. A circle is drawn with the point representing $$1$$ on the number line as the center and the length of the diagonal of the square as the radius, intersecting the negative half of the number line at point $$A$$. The number represented by point $$A$$ is ___.

1-\sqrt{2}

<image>As shown in the figure, the area of rectangle OABC is 18. Its diagonal OB intersects the hyperbola y=$\frac{k}{x}$ at point D, and OD:DB=2:1, then k=．

8

<image>As shown in the figure, O is a point on the straight line l, ∠1 + ∠2 = 78°42′, then ∠AOB =.

101°18′

<image>As shown in the figure, there is a square plastic template ABCD with a side length of 4. The right-angle vertex of a sufficiently large right-angled triangle is placed at point A, with its two legs intersecting CD at point F and the extension of CB at point E. The area of quadrilateral AECF is ______.

16

<image>As shown in the figure, water is transferred from bucket A to bucket B. Initially, bucket A has a liters of water. After t minutes, the remaining water y liters satisfies the function y=ae^{-nt}. Therefore, the water in bucket B is y=a-ae^{-nt}. Assuming that after 5 minutes, the water in bucket A and bucket B is equal, then after _____ minutes, the water in bucket A will be \frac{a}{8}L.

10

<image>As shown in the figure, triangle ABC is translated 3cm to the left to obtain triangle DEF, where points E, B, F, and C lie on the same straight line. If the perimeter of triangle ABC is 12cm, then the perimeter of quadrilateral ACED is cm.

18cm

<image>Fold the right figure into a cube, the maximum sum of the numbers on the opposite faces is ______.

9

<image>As shown in the figure, given $$AB\parallel DE$$, $$BC\parallel EF$$, then the number of pairs of homothetic triangles in the figure is ___.

4

<image>As shown in the figure, in $$\triangle ABC$$, point $$O$$ is the midpoint of $$BC$$. A line through point $$O$$ intersects the lines $$AB$$ and $$AC$$ at two distinct points $$M$$ and $$N$$, respectively. If $$\overrightarrow{AB}=m\overrightarrow{AM}$$ and $$\overrightarrow{AC}=n\overrightarrow{AN}$$, then the value of $$m+n$$ is ___.

2

<image>As shown in the figure, there is some water in a cubic container with an edge length of 10 cm, and the water height is 7 cm. A rectangular iron block, 8 cm in height, is placed vertically into the water (the base of the iron block is parallel to the bottom of the container). Before the iron block is completely submerged, the water is already full, and the submerged part of the iron block is 6 cm high. The volume of this iron block is ______ cubic centimeters.

400

<image>In the flowchart shown, the output value of $n$ is.

4

<image>As shown in the figure, the perimeter of ▱ABCD is 18 cm, and the diagonals AC and BD intersect at point O. If the difference in the perimeters of △AOD and △AOB is 5 cm, then the length of side AB is ______ cm.

2

<image>Place a small ball at the top entrance of the container shown in the figure, and the ball will fall freely. During its fall, the ball will encounter a black obstacle 3 times, and it will finally fall into bag A or bag B. It is known that each time the ball encounters a black obstacle, the probability of falling to the left or right is $$\dfrac{1}{2}$$, then the probability of the ball falling into bag A is ___.

\dfrac{3}{4}

<image>In the figure, in △ABC, AB = 8, AC = 6, AD and AE are the angle bisector and median respectively. A perpendicular CG is drawn from point C to AD at F, intersecting AB at G. EF is connected. Find the length of EF.

1

<image>Given y = , then the arithmetic square root of xy is ______.

6

<image>In triangle ABC, ∠ABC = 2∠C, AP and BQ are the angle bisectors of ∠BAC and ∠ABC, respectively. If the perimeter of triangle ABQ is 18, and BP = 4, then the length of AB is

7

<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ACB=90^{\circ}$$, point $$F$$ is on the extension of $$AC$$ such that $$CF=\dfrac {1}{2}AC$$, and $$DE$$ is the midsegment of $$\triangle ABC$$. If $$\angle 1=30^{\circ}$$ and $$DE=2$$, then the perimeter of quadrilateral $$AFED$$ is ___.

16

<image>As shown in the figure, the result output by the algorithm is ___.

21

<image>As shown in the figure, $$AB$$ is the diameter of $$\odot O$$, and chord $$CD$$ perpendicularly bisects $$OB$$ at point $$E$$. Connecting $$OD$$ and $$BC$$, if $$BC=1$$, then the area of sector $$OBD$$ is ___.

\dfrac{\pi }{6}

<image>As shown in the figure, the area of △ABC is 24, D is the midpoint of BC, and E is the midpoint of AC. What is the area of △CDE?

6

<image>If the following program flowchart is executed, then the output $S=$.

20

<image>As shown in the figure, in circle O, AB is the diameter, and point C is on circle O. The bisector of ∠ACB intersects circle O at D. Then ∠ABD = {{ }}°.

45

<image>The program in the figure calculates the function value. If the input value of x is $\frac{3}{2}$, then the output result y is.

0.5

<image>As shown in the figure, place right triangle ABC in a rectangular coordinate system, where ∠CAB = 90°, BC = 5, and the coordinates of points A and B are (﹣1, 0) and (﹣4, 0), respectively. Triangle ABC is translated to the left along the x-axis. When point C falls on the line y = -2x - 6, how many units has point C been translated to the left along the x-axis?

4

<image>As shown in the figure, line l is perpendicular to the x-axis at point P, and intersects the graphs of the inverse proportion functions y_1 = k_1 / x (x > 0) and y_2 = k_2 / x (x > 0) at points A and B, respectively. Connecting OA and OB, the area of triangle OAB is known to be 1. Then k_1 - k_2 =.

2

<image>The following is a statistical graph of Xiao Hua's scores in five math tests. The average score of Xiao Hua's five tests is ______ points.

92

<image>Among the following four car logos, the number of logos that are both centrally symmetric and axially symmetric is ______.

1

<image>During the flag-raising ceremony at the Chinese National Day parade, as shown in the figure, a row of seats and the flagpole are on the same vertical plane perpendicular to the ground on a viewing stand with an incline of 15°. The angles of elevation to the top of the flagpole from the first and last rows are 60° and 30°, respectively, and the distance between the first and last rows is 10\sqrt{6}m. What is the height of the flagpole?

30

<image>As shown in the figure, let $$G$$ be the centroid of $$ \triangle ABC$$, and let the midpoints of $$BC$$, $$CA$$, and $$AB$$ be $$D$$, $$E$$, and $$F$$, respectively. Then $$\overrightarrow{GD} + \overrightarrow{GE} + \overrightarrow{GF} = $$___.

\overrightarrow{0}

<image>In triangle ABC, it is known that AB = AC, BC = BD, AD = DE = EB, the measure of angle A is ______.

45°

<image>Given the power function $$f(x)=x^{a}$$ with some corresponding values as shown in the table below, then the solution set of the inequality $$f(|x|) \leqslant 2$$ is ___.

\{x|-4 \leqslant x \leqslant 4\}

<image>Determine whether the input number $$x$$ is positive. If it is, output its square; if not, output its opposite. The blank should be filled with ___.

x \leqslant 0

<image>In the figure, in $$\triangle ABC$$, $$AC=BC=10cm$$, $$\angle A=30\degree$$, $$CD$$ is the median of $$\triangle ABC$$, and a line through $$D$$ parallel to $$BC$$ intersects the bisector of $$\angle BCD$$ at point $$E$$. The length of $$DE$$ is ______.

5

<image>In the right triangle ABC, AC⊥BC, and CD⊥AB at D, AD=4, BD=2, then CD=．

2\sqrt{2}

<image>As shown in the figure, points $$P$$ and $$Q$$ are on the graph of the inverse proportion function $$y=\dfrac{k}{x}$$. $$PA \perp y$$-axis at point $$A$$, $$QN \perp x$$-axis at point $$N$$, $$PM \perp x$$-axis at point $$M$$, and $$QB \perp y$$-axis at point $$B$$. Connecting $$PB$$ and $$QM$$, if the area of $$\triangle ABP$$ is denoted as $$S_{1}$$, and the area of $$\triangle QMN$$ is denoted as $$S_{2}$$, then $$S_{1}$$ ___ $$S_{2}$$ (fill in "$$>$$", "$$<$$", or "$$=$$").

=

<image>As shown in the figure, given that the edge length of the cube $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ is a, the angle formed by the skew lines $B{{C}_{1}}$ and $AC$ is．

60{}^\circ

<image>As shown in the figure, the lines $$AD∥BE$$, $$AC$$ and $$BC$$ bisect $$∠BAD$$ and $$∠ABE$$ respectively, and $$∠CAD=55^{\circ}$$. Then $$∠CBE=$$ ___.

35^{\circ}

<image>As shown in the figure, a soybean is randomly scattered in a square with a side length of 1. The probability that it lands in the shaded area is.

\frac{1}{3}

<image>With the continuous advancement of spiritual civilization, citizens are spending more and more time each day on reading books, reading newspapers, and participating in 'National Fitness Campaigns.' The figure below is a frequency distribution histogram created by a reporter from our city's evening newspaper after a sample survey of some citizens' time spent on these activities. The sum of the areas of the first seven rectangles from left to right is $$\number{0.94}$$, and the frequency of the last group is $$12$$. What is the sample size of this survey?

200

<image>As shown in the figure, $OP$ bisects $\angle AOB$, $PE \perp AO$ at point $E$, $PF \perp BO$ at point $F$, and $PE = 6\text{cm}$. Then the distance from point $P$ to $OB$ is $\text{cm}$.

6