xyliu6/k12-freeform · Datasets at Hugging Face

problem stringlengths 14 1.27k	answer stringlengths 1 93	id stringlengths 6 9
<image> As shown in the figure, the chord $$CD$$ intersects the diameter $$AB$$ of circle $$\odot O$$. If $$\angle BAD=50^{ \circ }$$, then $$\angle ACD=$$___ degrees.	40	math_4438
<image> As shown in the figure, perform the calculation according to the program. If the input number is $-2$, then the output number is.	-50	math_245
<image> A rectangular paper sheet $$ABCD$$ is folded as shown in the figure, with $$EF$$ and $$EG$$ being the creases. What is the value of $$\angle AEF + \angle BEG$$ in degrees?	90	math_983
<image> As shown in the figure, the line y$_{1}$ = -x + 6 intersects with the curve y$_{2}$ = $\frac{8}{x}$ (x > 0). If -x + 6 > $\frac{8}{x}$, determine the range of values for the independent variable x.	2 < x < 4	math_5068
<image> Perform the operation according to the program shown in the figure: Rule: The program runs to the question 'Is the result greater than 65?' as one operation, and the operation stops after 4 runs. The number of integers x that can be input is ___.	4	math_1542
<image> As shown in the figure, lines $AB$ and $CD$ intersect at point $O$, and $EO \bot AB$. The foot of the perpendicular is $O$, and $\angle EOC = 35{}^\circ$. What is the measure of $\angle AOD$ in degrees?	125	math_4797
<image> As shown in the figure, two triangular rulers are stacked together. If $AB=4\text{cm}$, then ${S}_{\Delta ACF}=$$\text{cm}^2$.	2	math_4479
<image> As shown in the figure, if $OP$ bisects $\angle AOB$ and $OP \bot AB$, then the equal line segments in the figure are:	OA=OB, AP=BP	math_3536
<image> As shown in the figure, in rectangle $$ABCD$$, $$AC$$ and $$BD$$ intersect at point $$O$$, $$OE \bot AC$$ intersects $$BC$$ at point $$E$$, and $$AE$$ is connected. If $$AB=1$$ and $$BC=\sqrt{3}$$, then $$\dfrac{1}{\tan\angle EAO}=$$ ___.	\sqrt{3}	math_864
<image> Given the sequence $$\{a_n\}$$, where $$a_1=1$$ and $$a_{n+1}=2a_n+n-1$$, if the operation is performed using the program flowchart shown in the figure, then the output value of $$n$$ is ______.	11	math_881
<image> As shown in the figure, if the input is $a=1010$, $k=8$, $n=4$, then the output $b=$.	520	math_6045
<image> A square paper strip is folded as shown in the figure. If the resulting $$\angle AOB' = 40^{\circ}$$, then $$\angle B'OG =$$ ___ degrees.	70	math_7933
<image> BD is the median of \(\triangle ABC\), AB = 5, BC = 3, the difference in the perimeters of \(\triangle ABD\) and \(\triangle BCD\) is ___.	2	math_5283
<image> As shown in the figure, points A and B are separated by a pond. Choose any point C outside AB, and connect AC and BC. Take points M and N on AC and BC respectively, such that they are the points closer to C that divide AC and BC into three equal parts. The length of MN is measured to be 38 meters. What is the length of AB in meters?	114	math_6300
<image> As shown in the figure, the side length of square $$ABCD$$ is $$4$$. The diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Point $$E$$ is on the extension of side $$DC$$. If $$∠CAE=15^{\circ}$$, then $$AE=$$ ___.	8	math_5717
<image> As shown in the figure, given AB∥CD∥EF, BC∥AD, and AC bisects ∠BAD, how many angles in the figure are equal to ∠AGE?	5	math_5393
<image> In triangle ABC, point D is on side AB, and AB = 4AD. Let $\overrightarrow{\text{AB}}=\overrightarrow{a}$ and $\overrightarrow{\text{AC}}=\overrightarrow{b}$. Express vector $\overrightarrow{\text{DC}}$ in terms of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.	-\frac{1}{4}\vec{a}+\vec{b}	math_2903
<image> As shown in the figure, $AB$ is the diameter, points $C$ and $D$ are on the circle. If $\angle CAB=25{}^\circ $, then the measure of $\angle ADC$ is ___ degrees?	65	math_7035
<image> Below is the pseudocode of an algorithm. When this algorithm is executed, the output result is ___.	3	math_1088
<image> As shown in the figure, in the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, a plane passing through the midpoint $$E$$ of $$BB_{1}$$ is parallel to the plane $$ACB_{1}$$ and intersects $$AB$$ at $$M$$ and $$BC$$ at $$N$$. Then $$MN = $$___$$AC$$.	\dfrac{1}{2}	math_1043
<image> After the following program runs, the output result is _____.	22,-22	math_1471
<image> As shown in the figure, in $$\triangle ABC$$, $$AB=7\ \unit{cm}$$, $$AC=4\ \unit{cm}$$, the perpendicular bisector of $$BC$$ intersects $$AB$$ and $$BC$$ at points $$D$$ and $$E$$, respectively. The perimeter of $$\triangle ACD$$ is ___ $$\unit{cm}$$.	11	math_3439
<image> Execute the program flowchart shown in the figure, the output result is.	(-4,0)	math_7355
<image> In the right triangle ABC, ∠B = 90°, BC = 4, AC = 5. Triangle ABC is folded so that point C coincides with point A, resulting in the crease DE. The perimeter of triangle ABE is equal to.	7	math_5371
<image> In the figure, in $\vartriangle ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively, and point $F$ is on $BC$. $ED$ is the angle bisector of $\angle AEF$. If $\angle C = 80^\circ$, then the measure of $\angle EFB$ is ___ degrees?	100	math_3840
<image> Execute the pseudocode shown in the figure, the output result is .	10	math_7545
<image> As shown in the figure, $$AB$$ and $$CD$$ intersect at point $$O$$, $$∠AOD + ∠BOC = 280^{\circ}$$, then $$∠AOC =$$ ___ degrees.	40	math_979
<image> Look at the picture and write the number, write as:______.	23	math_7431
<image> A journalist conducted a sample survey on the time (in minutes) some students at a school spent reading during the holiday and drew a frequency distribution histogram. The sum of the frequencies corresponding to the first 5 rectangles from left to right is 0.9, and the frequency of the last group is 15. The number of people surveyed in this sample is ___ people. (Note: On the horizontal axis, each group of data includes the minimum value but does not include the maximum value)	150	math_661
<image> As shown in the figure, the three vertices of $$\triangle ABC$$ and an internal point $$P_1$$ divide $$\triangle ABC$$ into $$3$$ non-overlapping smaller triangles; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$ divide $$\triangle ABC$$ into $$5$$ non-overlapping smaller triangles; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$, $$P_3$$ divide $$\triangle ABC$$ into $$7$$ non-overlapping smaller triangles; $$\cdots$$; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$, $$P_3$$, $$\cdots$$, $$P_n$$ divide $$\triangle ABC$$ into ______ non-overlapping smaller triangles.	2n+1	math_2046
<image> The histogram below shows the frequency distribution of scores for 48 students in a math test (scores are integers only). The height ratio of the rectangles from left to right is 1:3:6:4:2. According to the graph, the number of students with scores in the range of 70.5 to 80.5 is .	18	math_8070
<image> Figure 1 is an electronic photo frame on Xiao Zhi's desk. Its side is abstracted into the geometric shape shown in Figure 2. Given that $$AB=AC=\quantity{15}{cm}$$ and $$\angle BAC=40\unit{^{ \circ }}$$, what is the distance from point $$A$$ to $$BC$$ in cm (reference data: $$\sin 20\unit{^{ \circ }}\approx 0.342$$, $$\cos 20\unit{^{ \circ }}\approx 0.940$$, $$\sin 40\unit{^{ \circ }}\approx 0.643$$, $$\cos 40\unit{^{ \circ }}\approx \number{0.766}$$. The result should be accurate to $$\quantity{0.1}{cm}$$, and a scientific calculator can be used)?	14.1	math_7954
<image> As shown in the figure, $DE$ is the midline of $\Delta ABC$, then the ratio of the area of $\vartriangle ADE$ to the area of quadrilateral $DBCE$ is.	\frac{1}{3}	math_4497
<image> The partial graph of the parabola $$y=-x^2+bx+c$$ is shown in the figure. If $$y>0$$, then the range of $$x$$ is ______.	-3<x<1	math_901
<image> As shown in the figure, point $$E$$ is the midpoint of side $$BC$$ of square $$ABCD$$. If a point is randomly selected within the square, the probability that it lies in the shaded area is ___.	\dfrac{1}{4}	math_6905
<image> As shown in the figure, in a grid paper composed of squares with side length $$1$$, forming a $$4 \times 2$$ grid, how many different directions are there among all vectors with their endpoints at the grid points and with a magnitude of $$\sqrt{5}$$?	8	math_3035
<image> As shown in the figure, $$\odot O$$ is the incircle of $$\triangle ABC$$, and the points of tangency are $$D$$, $$E$$, and $$F$$, with $$BD=3$$ and $$AE=2$$. Then, $$AB=$$______．	5	math_3616
<image> As shown in the figure, a cylindrical measuring cup with a base radius of $$R$$ is partially filled with water. When a solid sphere with radius $$r$$ is placed in the cup, it is completely submerged, and the water level rises exactly by $$r$$. Then, $$\dfrac{R}{r}=$$ ___.	\dfrac{2\sqrt{3}}{3}	math_4842
<image> As shown in the figure, in $$\triangle ABC$$, $$M$$ and $$N$$ are the midpoints of $$AC$$ and $$BC$$, respectively. If $$S _{\triangle CMN}=1$$, then $$S_{quadrilateral ABNM}=$$____.	3	math_6489
<image> In an experiment, four sets of values for $$\left (x, y \right ) $$ were measured as . According to the table, the regression equation is $$\hat{y}=-5x+\hat{a}$$, and using this model, the predicted value of $$y$$ when $$x$$ is $$20$$ is ___.	26.5	math_8011
<image> As shown in the figure, it is known that $$AC \bot BC$$ at point $$C$$, $$AD \bot BD$$ at point $$D$$, and $$AC = AD$$. If $$∠ACE = 25^{\circ}$$, then the measure of $$∠ADE$$ is ___ degrees.	25	math_899
<image> As shown in the figure, in the rectangular prism $$ABCD - A_{1}B_{1}C_{1}D_{1}$$, $$BC = 2$$, $$AA_{1} = 1$$, and points $$E$$ and $$F$$ are on $$AD$$ and $$BC$$ respectively, with $$EF \parallel AB$$. If the dihedral angle $$C_{1} - EF - C$$ is $$45^{\circ}$$, then $$BF = $$___.	1	math_3032
<image> As shown in the figure, in rectangle $$ABCD$$, $$AB=6$$. The first translation moves rectangle $$ABCD$$ 5 units to the right along $$AB$$, resulting in rectangle $$A_{1}B_{1}C_{1}D_{1}$$. The second translation moves rectangle $$A_{1}B_{1}C_{1}D_{1}$$ 5 units to the right along $$A_{1}B_{1}$$, resulting in rectangle $$A_{2}B_{2}C_{2}D_{2}$$, and so on. The $$n$$-th translation moves rectangle $$A_{n-1}B_{n-1}C_{n-1}D_{n-1}$$ 5 units to the right along $$A_{n-1}B_{n-1}$$, resulting in rectangle $$A_{n}B_{n}C_{n}D_{n} (n > 2)$$. If the length of $$AB_{n}$$ is 56, then $$n$$ is ___.	10	math_6943
<image> As shown in the figure, ∠DAB = ∠EAC = 65°, AB = AD, AC = AE, BE and CD intersect at point O, AB and CD intersect at P, AC and BE intersect at F. What is the measure of ∠DOE in degrees?	115	math_272
<image> As shown in the figure, $$\angle A=98^{\circ}$$, $$\angle B=105^{\circ}$$, $$\angle C=75^{\circ}$$, then $$\angle D=$$ ___ degrees.	82	math_5422
<image> The pie chart below shows the allocation of Lelé's daily activities. If Lelé wishes to adjust his daily reading time to 2 hours, by how many minutes does Lelé need to increase his reading time?	60	math_5020
<image> As shown in the figure, a rectangular piece of paper ABCD is folded along AE upwards, so that point B lands on point F on side DC. If the perimeter of △AFD is 9, and the perimeter of △ECF is 3, then AB + BC =.	6	math_6965
<image> Execute the program flowchart shown in the figure, the output value of $$i$$ is ___.	4	math_6388
<image> As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, with the radius of $\odot O$ being $R=2$, and $\text{sin}B=\frac{3}{4}$. Find the length of the chord $AC$.	3	math_5494
<image> As shown in the figure, points A and B are on the number line, corresponding to the numbers -4 and $\frac{2\text{x}+2}{3\text{x-}5}$, respectively. Given that the distances from points A and B to the origin are equal, find x.	\frac{11}{5}	math_7826
<image> As shown in the figure, in trapezoid $$ABCD$$, $$AB \parallel CD$$, $$S_{\triangle DEC}:S_{\triangle CEB}=1:2$$. Then, $$S_{\triangle DEC}:S_{\triangle EAB}=$$ ___.	1:4	math_1330
<image> The graph of the quadratic function $y=ax^2+bx$ is shown in the figure. If the quadratic equation $ax^2+bx+m=0$ has real roots, then the maximum value of $m$ is	3	math_6499
<image> Xiao Lin, Xiao Fang, and Xiao Liang are playing a dart game, each throwing 5 darts. The rules state that scoring is the same for darts landing in the same ring. The target and scoring situation are shown in the figure. What is Xiao Liang's score in points?	21	math_2018
<image> As shown in the figure, the lines $$a \parallel b$$, $$\angle 1=125^{ \circ }$$. What is the measure of $$\angle 2$$ in degrees?	55	math_6458
<image> Read the following pseudocode. If the algorithm outputs the result $$-5$$, then the initial value $$x$$ of $$a$$ is ___.	-1	math_2984
<image> As shown in the figure, in $$\triangle ABC$$, $$D$$ is the midpoint of side $$BC$$, $$\overrightarrow{BA}=\overrightarrow{a}$$, $$\overrightarrow{BC}=\overrightarrow{b}$$, then $$\overrightarrow{DA}=$$___.	\overrightarrow{a}-\dfrac{1}{2}\overrightarrow{b}	math_2842
<image> As shown in the figure, it is a flowchart of an algorithm. When the input value of $$x$$ is $$5$$, the output result is ___.	2	math_6980
<image> Eight small cubes of the same size are glued together to form a larger cube as shown in Figure 1. The three views of the resulting geometric shape are shown in Figure 2. If Xiao Ming removes several of the eight small cubes, keeping the remaining small cubes in their original positions, and ensures that the three views of the new geometric shape are still as shown in Figure 2, then the maximum number of small cubes he can remove is ______.	4	math_6235
<image> In the Cartesian coordinate system, the vertices of the square $ABCD$ have coordinates $A$(1, 1), $B$(1, -1), $C$(-1, -1), $D$(-1, 1), and there is a point $P$(0, 2) on the $y$-axis. Construct the point $P_1$ symmetric to point $P$ with respect to point $A$, the point $P_2$ symmetric to point $P_1$ with respect to point $B$, the point $P_3$ symmetric to point $P_2$ with respect to point $C$, the point $P_4$ symmetric to point $P_3$ with respect to point $D$, the point $P_5$ symmetric to point $P_4$ with respect to point $A$, and the point $P_6$ symmetric to point $P_5$ with respect to point $B$, and so on. What are the coordinates of $P_{2020}$?	(0, 2)	math_1375
<image> The oblique axonometric drawing of a horizontally placed triangle is an isosceles right triangle A′B′O′. If O′B′ = 1, then what is the area of the original △ABO?	\sqrt{2}	math_2898
<image> As shown in the figure, the perimeter of rhombus ABCD is 40 cm, and the length of the diagonal AC is 10 cm. What are the measures of the adjacent interior angles of the rhombus in degrees?	60, 120	math_6701
<image> As shown in the figure, a right-angled triangle with a $$45^{\circ}$$ angle has two vertices placed on the opposite edges of a ruler. If $$\angle 1=20^{\circ}$$, then the degree measure of $$\angle 2$$ is ___ degrees.	25	math_5382
<image> As shown in the figure, in △ABC, BC = 5 cm; △ABC is translated along the line containing BC to △DEF (see figure); if EC = 2 cm, then CF = cm.	3	math_1153
<image> As shown in the figure, Uncle Wu has a piece of land in the shape of an equilateral triangle ABC. It is known that E and F are the midpoints of sides AB and AC, respectively, and EF measures 5 meters. He wants to enclose the quadrilateral BCFE with a fence to raise chicks. What is the length of the fence needed in meters?	25	math_6091
<image> As shown in the figure, Xiao Ming stands at point $$C$$ to observe points $$A$$ and $$E$$, the top points of buildings Jia and Yi, respectively. Points $$C$$, $$E$$, and $$A$$ are collinear, and points $$B$$ and $$D$$ are directly below points $$E$$ and $$A$$, respectively, with points $$D$$, $$B$$, and $$C$$ also collinear. If the distance between point $$B$$ and point $$C$$ is $$\quantity{20}{m}$$, the distance between point $$D$$ and point $$C$$ is $$\quantity{40}{m}$$, and the height of building Yi $$BE$$ is $$\quantity{15}{m}$$, then the height of building Jia $$AD$$ is ___$$\unit{m}$$ (Xiao Ming's height is negligible).	30	math_1198
<image> Execute the following pseudocode, the output value of $$x$$ is ___.	25	math_1373
<image> The figure below shows the frequency distribution histogram of the speeds of cars passing through a certain speed monitoring point on National Highway 204. Among the 300 cars passing through this monitoring point, approximately how many cars have speeds in the range [60, 80)?	150	math_6987
<image> In response to the 'Red Songs Resound in China' campaign, a town held a 'Red Song' singing competition. The organizing committee stipulates: any participant's score $$x$$ must satisfy: $$60 \leqslant x < 100$$. After the competition, the scores of all participants were organized as shown in Table (1). According to the information provided in Table (1), $$n=$$______.	0.3	math_1990
<image> The graph of the function $$y=f(x)$$ is shown below. Which of the following statements is correct? 1. $$f'(3) > 0$$; 2. $$f'(3) < 0$$; 3. $$f'(3)=0$$; 4. The sign of $$f'(3)$$ is uncertain.	2	math_5320
<image> As shown in the figure, the graph of the function $$f(x)$$ is the line segment $$ABC$$, where the coordinates of points $$A$$, $$B$$, and $$C$$ are $$(0,4)$$, $$(2,0)$$, and $$(6,4)$$, respectively. Then $$f(f(f(2)))=$$ ___.	2	math_5786
<image> As shown in the figure, if the parabola $y=ax^2+bx+c$ has no intersection points with the $x$-axis, then $a, b, c$ should satisfy the relationship:	b^2-4ac < 0	math_6394
<image> After the computer executes the following pseudocode, if the input is $$a=2$$, $$b=3$$, what is the output?	6	math_2017
<image> As shown in the figure, after running the pseudocode of the algorithm, the value of $$S$$ that is output is ___.	17	math_4135
<image> The figure below has ( ______ ) lines of symmetry.	1	math_7282
<image> In the 'Chinese Classics Recitation' activity, a school conducted a random survey of 50 students to understand the extracurricular reading situation of the entire school's 1300 students. The survey collected data on the students' weekly extracurricular reading time and created the bar chart shown in the figure. Based on the data in the chart, estimate the number of students among the 1300 who read for at least 7 hours per week.	520	math_4417
<image> As shown in the figure, the curve $$C$$: $$y=\sqrt{x}$$, suppose the line $$l_{1}$$ is tangent to the curve $$C$$ at point $$P$$, and the line $$l_{2}$$ passes through point $$P$$ and is perpendicular to line $$l_{1}$$. If the line $$l_{2}$$ intersects the $$x$$-axis at point $$Q$$, and the projection of point $$P$$ on the $$x$$-axis is point $$R$$, then $$\left \lvert RQ\right \rvert =$$___.	\dfrac{1}{2}	math_6052
<image> As shown in the figure, in △ABC, ∠ACB = 90°, ∠B = 30°, CD is perpendicular to AB at point D, AB = 4, then the length of BD is.	3	math_6815
<image> The following is a statistical chart based on the donation situation of a middle school for earthquake-stricken areas. It is known that the school has 3000 students. Calculate the total amount of donations made by the school.	37770	math_5505
<image> Xiaoliang conducted a questionnaire survey on water-saving methods among 60 classmates (each choosing one method). The number of people is shown in the chart. If this data is represented in a pie chart, what is the central angle of the sector representing 'Multiple Uses of Water' in degrees?	240	math_4743
<image> Given that the maximum speed limit on a certain road is $$\quantity{50}{km/h}$$, the electronic monitoring system measures the speeds of 6 consecutive cars as shown in the figure. If 2 cars are randomly selected, the probability that exactly 1 car is speeding is ___.	\dfrac{8}{15}	math_7878
<image> There are four game boards. If a soybean falls on the shaded area, a prize can be won. Xiao Ming hopes to win a prize. He should choose the game board __ (fill in the number).	1	math_1421
<image> As shown in the figure, the function $$f(x)=x^{n^{2}-2n-3}(n \in \mathbf{Z})$$ is illustrated. What is the value of $$n$$?	1	math_5478
<image> As shown in the figure, $$AB\parallel CD$$, $$PB$$ bisects $$\angle ABC$$, $$PC$$ bisects $$\angle DCB$$. Then $$\angle P=$$ ___ degrees.	90	math_2
<image> As shown in the figure, △ABC is inscribed in circle ⊙O, BD is the diameter of ⊙O, ∠CBD = 21°, then the measure of ∠A is ___ degrees?	69	math_3551
<image> Given the graph of the linear function $y=kx+b$ as shown, the range of $y$ when $x < 0$ is.	y < -2	math_2467
<image> As shown in the figure, in rectangle ABCD, BC=4, AB=$2\sqrt{3}$, and P is a point on the extension of AD. ${{S}_{\Delta PBC}}=$ ___?	4 \sqrt{3}	math_3835
<image> Teacher Li copied the probability distribution of a random variable $$\xi$$ from a textbook as shown in the table: Student Wang was asked to calculate the mathematical expectation of $$\xi$$. Although the value at the "?" is completely illegible and the two "!"s are blurry, it can be determined that the two "!"s have the same value. Based on this, Student Wang correctly gave the answer $$E \xi =$$___.	2	math_6585
<image> On April 20, 2013, a major earthquake of magnitude 7.0 struck Lushan County in Ya'an City. The people of the entire Sichuan province united to respond to the disaster. A class organized a 'Donate Pocket Money, Show Love' activity. The donation situation of the 50 students in the class is shown in the figure below. What is the mode of the donation amounts?	20	math_707
<image> In the right triangle △ABC, ∠BAC=90°. Triangle ABC is rotated clockwise around point A to form triangle ADE, with point B's corresponding point D landing exactly on BC. If AC=4$\sqrt{3}$ and ∠B=60°, then the length of CD is	4	math_3085
<image> The graph of an inverse proportion function in the third quadrant is shown in the figure. A is any point on the graph, AM is perpendicular to the x-axis at point M, and O is the origin. If the area of △AOM is 3, then the expression of this inverse proportion function is ______.	y = 6/x	math_234
<image> If the side length of a regular hexagon is 3, then the length of its longer diagonal is.	6	math_4091
<image> In the figure, point O is a point inside △ABC, and the distances from point O to the three sides AB, BC, and CA are equal. Connecting OB and OC, if ∠A = 78°, then the degree measure of ∠BOC is ___ degrees.	129	math_1250
<image> Lelé found that three balls of the same size can fit perfectly inside a cylindrical box (the thickness of the bottom and lid is negligible), as shown in the figure. What fraction of the total volume of the box is occupied by the three balls? (The volume of a sphere is given by the formula $V=\frac{4}{3}\pi r^3$)	\frac{2}{3}	math_3157
<image> A research team obtained a set of data in an experiment and organized it into the scatter plot shown below. Among the following functions, which one best approximates the relationship between y and t? (Fill in the sequence number). 1. y = 2^t; 2. y = 2t^2; 3. y = t^3; 4. y = log_2 t.	4	math_5512
<image> The perspective view of a geometric solid is shown in the figure. Among the following four top views, which one is correct?	2	math_90
<image> In the figure, in rectangle ABCD, BE is perpendicular to AC, intersecting AC and AD at points F and E, respectively. Given AF = 2 and AC = 6, what is the length of AB?	2 \sqrt{3}	math_3803
<image> In the figure, $$\triangle ABC$$, $$DE$$ is the perpendicular bisector of $$AC$$, $$AE = \quantity{6}{cm}$$, and the perimeter of $$\triangle ABD$$ is $$\quantity{26}{cm}$$, then the perimeter of $$\triangle ABC$$ is ___$$\unit{cm}$$.	38	math_2884
<image> As shown in the figure, in $\vartriangle ABC$, $\angle B = \angle C = 60{}^\circ$, point $D$ is on side $AB$, $DE \perp AB$, and intersects side $AC$ at point $E$. If $AD = 1$ and $BC = 6$, then $CE =$．	4	math_241
<image> As shown in the figure, in the pentahedron $$ABCDEF$$, $$FA \perp$$ plane $$ABCD$$, $$AD \parallel BC \parallel FE$$, $$AB \perp AD$$, $$AF=AB=BC=FE=\dfrac{1}{2}AD$$. What is the angle between the skew lines $$BF$$ and $$ED$$ in degrees?	60	math_4005
<image> As shown in the figure, if the input number is "$$37$$", then the output result is ___.	73	math_4146

<image> As shown in the figure, the chord $$CD$$ intersects the diameter $$AB$$ of circle $$\odot O$$. If $$\angle BAD=50^{ \circ }$$, then $$\angle ACD=$$___ degrees.

40

math_4438

<image> As shown in the figure, perform the calculation according to the program. If the input number is $-2$, then the output number is.

-50

math_245

<image> A rectangular paper sheet $$ABCD$$ is folded as shown in the figure, with $$EF$$ and $$EG$$ being the creases. What is the value of $$\angle AEF + \angle BEG$$ in degrees?

90

math_983

<image> As shown in the figure, the line y$_{1}$ = -x + 6 intersects with the curve y$_{2}$ = $\frac{8}{x}$ (x > 0). If -x + 6 > $\frac{8}{x}$, determine the range of values for the independent variable x.

2 < x < 4

math_5068

<image> Perform the operation according to the program shown in the figure: Rule: The program runs to the question 'Is the result greater than 65?' as one operation, and the operation stops after 4 runs. The number of integers x that can be input is ___.

4

math_1542

<image> As shown in the figure, lines $AB$ and $CD$ intersect at point $O$, and $EO \bot AB$. The foot of the perpendicular is $O$, and $\angle EOC = 35{}^\circ$. What is the measure of $\angle AOD$ in degrees?

125

math_4797

<image> As shown in the figure, two triangular rulers are stacked together. If $AB=4\text{cm}$, then ${S}_{\Delta ACF}=$$\text{cm}^2$.

2

math_4479

<image> As shown in the figure, if $OP$ bisects $\angle AOB$ and $OP \bot AB$, then the equal line segments in the figure are:

OA=OB, AP=BP

math_3536

<image> As shown in the figure, in rectangle $$ABCD$$, $$AC$$ and $$BD$$ intersect at point $$O$$, $$OE \bot AC$$ intersects $$BC$$ at point $$E$$, and $$AE$$ is connected. If $$AB=1$$ and $$BC=\sqrt{3}$$, then $$\dfrac{1}{\tan\angle EAO}=$$ ___.

\sqrt{3}

math_864

<image> Given the sequence $$\{a_n\}$$, where $$a_1=1$$ and $$a_{n+1}=2a_n+n-1$$, if the operation is performed using the program flowchart shown in the figure, then the output value of $$n$$ is ______.

11

math_881

<image> As shown in the figure, if the input is $a=1010$, $k=8$, $n=4$, then the output $b=$.

520

math_6045

<image> A square paper strip is folded as shown in the figure. If the resulting $$\angle AOB' = 40^{\circ}$$, then $$\angle B'OG =$$ ___ degrees.

70

math_7933

<image> BD is the median of $\triangle ABC$, AB = 5, BC = 3, the difference in the perimeters of $\triangle ABD$ and $\triangle BCD$ is ___.

2

math_5283

<image> As shown in the figure, points A and B are separated by a pond. Choose any point C outside AB, and connect AC and BC. Take points M and N on AC and BC respectively, such that they are the points closer to C that divide AC and BC into three equal parts. The length of MN is measured to be 38 meters. What is the length of AB in meters?

114

math_6300

<image> As shown in the figure, the side length of square $$ABCD$$ is $$4$$. The diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Point $$E$$ is on the extension of side $$DC$$. If $$∠CAE=15^{\circ}$$, then $$AE=$$ ___.

8

math_5717

<image> As shown in the figure, given AB∥CD∥EF, BC∥AD, and AC bisects ∠BAD, how many angles in the figure are equal to ∠AGE?

5

math_5393

<image> In triangle ABC, point D is on side AB, and AB = 4AD. Let $\overrightarrow{\text{AB}}=\overrightarrow{a}$ and $\overrightarrow{\text{AC}}=\overrightarrow{b}$. Express vector $\overrightarrow{\text{DC}}$ in terms of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

-\frac{1}{4}\vec{a}+\vec{b}

math_2903

<image> As shown in the figure, $AB$ is the diameter, points $C$ and $D$ are on the circle. If $\angle CAB=25{}^\circ $, then the measure of $\angle ADC$ is ___ degrees?

65

math_7035

<image> Below is the pseudocode of an algorithm. When this algorithm is executed, the output result is ___.

3

math_1088

<image> As shown in the figure, in the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, a plane passing through the midpoint $$E$$ of $$BB_{1}$$ is parallel to the plane $$ACB_{1}$$ and intersects $$AB$$ at $$M$$ and $$BC$$ at $$N$$. Then $$MN = $$___$$AC$$.

\dfrac{1}{2}

math_1043

<image> After the following program runs, the output result is _____.

22,-22

math_1471

<image> As shown in the figure, in $$\triangle ABC$$, $$AB=7\ \unit{cm}$$, $$AC=4\ \unit{cm}$$, the perpendicular bisector of $$BC$$ intersects $$AB$$ and $$BC$$ at points $$D$$ and $$E$$, respectively. The perimeter of $$\triangle ACD$$ is ___ $$\unit{cm}$$.

11

math_3439

<image> Execute the program flowchart shown in the figure, the output result is.

(-4,0)

math_7355

<image> In the right triangle ABC, ∠B = 90°, BC = 4, AC = 5. Triangle ABC is folded so that point C coincides with point A, resulting in the crease DE. The perimeter of triangle ABE is equal to.

7

math_5371

<image> In the figure, in $\vartriangle ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively, and point $F$ is on $BC$. $ED$ is the angle bisector of $\angle AEF$. If $\angle C = 80^\circ$, then the measure of $\angle EFB$ is ___ degrees?

100

math_3840

<image> Execute the pseudocode shown in the figure, the output result is .

10

math_7545

<image> As shown in the figure, $$AB$$ and $$CD$$ intersect at point $$O$$, $$∠AOD + ∠BOC = 280^{\circ}$$, then $$∠AOC =$$ ___ degrees.

40

math_979

<image> Look at the picture and write the number, write as:______.

23

math_7431

<image> A journalist conducted a sample survey on the time (in minutes) some students at a school spent reading during the holiday and drew a frequency distribution histogram. The sum of the frequencies corresponding to the first 5 rectangles from left to right is 0.9, and the frequency of the last group is 15. The number of people surveyed in this sample is ___ people. (Note: On the horizontal axis, each group of data includes the minimum value but does not include the maximum value)

150

math_661

<image> As shown in the figure, the three vertices of $$\triangle ABC$$ and an internal point $$P_1$$ divide $$\triangle ABC$$ into $$3$$ non-overlapping smaller triangles; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$ divide $$\triangle ABC$$ into $$5$$ non-overlapping smaller triangles; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$, $$P_3$$ divide $$\triangle ABC$$ into $$7$$ non-overlapping smaller triangles; $$\cdots$$; the three vertices of $$\triangle ABC$$ and the internal points $$P_1$$, $$P_2$$, $$P_3$$, $$\cdots$$, $$P_n$$ divide $$\triangle ABC$$ into ______ non-overlapping smaller triangles.

2n+1

math_2046

<image> The histogram below shows the frequency distribution of scores for 48 students in a math test (scores are integers only). The height ratio of the rectangles from left to right is 1:3:6:4:2. According to the graph, the number of students with scores in the range of 70.5 to 80.5 is .

18

math_8070

<image> Figure 1 is an electronic photo frame on Xiao Zhi's desk. Its side is abstracted into the geometric shape shown in Figure 2. Given that $$AB=AC=\quantity{15}{cm}$$ and $$\angle BAC=40\unit{^{ \circ }}$$, what is the distance from point $$A$$ to $$BC$$ in cm (reference data: $$\sin 20\unit{^{ \circ }}\approx 0.342$$, $$\cos 20\unit{^{ \circ }}\approx 0.940$$, $$\sin 40\unit{^{ \circ }}\approx 0.643$$, $$\cos 40\unit{^{ \circ }}\approx \number{0.766}$$. The result should be accurate to $$\quantity{0.1}{cm}$$, and a scientific calculator can be used)?

14.1

math_7954

<image> As shown in the figure, $DE$ is the midline of $\Delta ABC$, then the ratio of the area of $\vartriangle ADE$ to the area of quadrilateral $DBCE$ is.

\frac{1}{3}

math_4497

<image> The partial graph of the parabola $$y=-x^2+bx+c$$ is shown in the figure. If $$y>0$$, then the range of $$x$$ is ______.

-3<x<1

math_901

<image> As shown in the figure, point $$E$$ is the midpoint of side $$BC$$ of square $$ABCD$$. If a point is randomly selected within the square, the probability that it lies in the shaded area is ___.

\dfrac{1}{4}

math_6905

<image> As shown in the figure, in a grid paper composed of squares with side length $$1$$, forming a $$4 \times 2$$ grid, how many different directions are there among all vectors with their endpoints at the grid points and with a magnitude of $$\sqrt{5}$$?

8

math_3035

<image> As shown in the figure, $$\odot O$$ is the incircle of $$\triangle ABC$$, and the points of tangency are $$D$$, $$E$$, and $$F$$, with $$BD=3$$ and $$AE=2$$. Then, $$AB=$$______．

5

math_3616

<image> As shown in the figure, a cylindrical measuring cup with a base radius of $$R$$ is partially filled with water. When a solid sphere with radius $$r$$ is placed in the cup, it is completely submerged, and the water level rises exactly by $$r$$. Then, $$\dfrac{R}{r}=$$ ___.

\dfrac{2\sqrt{3}}{3}

math_4842

<image> As shown in the figure, in $$\triangle ABC$$, $$M$$ and $$N$$ are the midpoints of $$AC$$ and $$BC$$, respectively. If $$S _{\triangle CMN}=1$$, then $$S_{quadrilateral ABNM}=$$____.

3

math_6489

<image> In an experiment, four sets of values for $$\left (x, y \right ) $$ were measured as . According to the table, the regression equation is $$\hat{y}=-5x+\hat{a}$$, and using this model, the predicted value of $$y$$ when $$x$$ is $$20$$ is ___.

26.5

math_8011

<image> As shown in the figure, it is known that $$AC \bot BC$$ at point $$C$$, $$AD \bot BD$$ at point $$D$$, and $$AC = AD$$. If $$∠ACE = 25^{\circ}$$, then the measure of $$∠ADE$$ is ___ degrees.

25

math_899

<image> As shown in the figure, in the rectangular prism $$ABCD - A_{1}B_{1}C_{1}D_{1}$$, $$BC = 2$$, $$AA_{1} = 1$$, and points $$E$$ and $$F$$ are on $$AD$$ and $$BC$$ respectively, with $$EF \parallel AB$$. If the dihedral angle $$C_{1} - EF - C$$ is $$45^{\circ}$$, then $$BF = $$___.

1

math_3032

<image> As shown in the figure, in rectangle $$ABCD$$, $$AB=6$$. The first translation moves rectangle $$ABCD$$ 5 units to the right along $$AB$$, resulting in rectangle $$A_{1}B_{1}C_{1}D_{1}$$. The second translation moves rectangle $$A_{1}B_{1}C_{1}D_{1}$$ 5 units to the right along $$A_{1}B_{1}$$, resulting in rectangle $$A_{2}B_{2}C_{2}D_{2}$$, and so on. The $$n$$-th translation moves rectangle $$A_{n-1}B_{n-1}C_{n-1}D_{n-1}$$ 5 units to the right along $$A_{n-1}B_{n-1}$$, resulting in rectangle $$A_{n}B_{n}C_{n}D_{n} (n > 2)$$. If the length of $$AB_{n}$$ is 56, then $$n$$ is ___.

10

math_6943

<image> As shown in the figure, ∠DAB = ∠EAC = 65°, AB = AD, AC = AE, BE and CD intersect at point O, AB and CD intersect at P, AC and BE intersect at F. What is the measure of ∠DOE in degrees?

115

math_272

<image> As shown in the figure, $$\angle A=98^{\circ}$$, $$\angle B=105^{\circ}$$, $$\angle C=75^{\circ}$$, then $$\angle D=$$ ___ degrees.

82

math_5422

<image> The pie chart below shows the allocation of Lelé's daily activities. If Lelé wishes to adjust his daily reading time to 2 hours, by how many minutes does Lelé need to increase his reading time?

60

math_5020

<image> As shown in the figure, a rectangular piece of paper ABCD is folded along AE upwards, so that point B lands on point F on side DC. If the perimeter of △AFD is 9, and the perimeter of △ECF is 3, then AB + BC =.

6

math_6965

<image> Execute the program flowchart shown in the figure, the output value of $$i$$ is ___.

4

math_6388

<image> As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, with the radius of $\odot O$ being $R=2$, and $\text{sin}B=\frac{3}{4}$. Find the length of the chord $AC$.

3

math_5494

<image> As shown in the figure, points A and B are on the number line, corresponding to the numbers -4 and $\frac{2\text{x}+2}{3\text{x-}5}$, respectively. Given that the distances from points A and B to the origin are equal, find x.

\frac{11}{5}

math_7826

<image> As shown in the figure, in trapezoid $$ABCD$$, $$AB \parallel CD$$, $$S_{\triangle DEC}:S_{\triangle CEB}=1:2$$. Then, $$S_{\triangle DEC}:S_{\triangle EAB}=$$ ___.

1:4

math_1330

<image> The graph of the quadratic function $y=ax^2+bx$ is shown in the figure. If the quadratic equation $ax^2+bx+m=0$ has real roots, then the maximum value of $m$ is

3

math_6499

<image> Xiao Lin, Xiao Fang, and Xiao Liang are playing a dart game, each throwing 5 darts. The rules state that scoring is the same for darts landing in the same ring. The target and scoring situation are shown in the figure. What is Xiao Liang's score in points?

21

math_2018

<image> As shown in the figure, the lines $$a \parallel b$$, $$\angle 1=125^{ \circ }$$. What is the measure of $$\angle 2$$ in degrees?

55

math_6458

<image> Read the following pseudocode. If the algorithm outputs the result $$-5$$, then the initial value $$x$$ of $$a$$ is ___.

-1

math_2984

<image> As shown in the figure, in $$\triangle ABC$$, $$D$$ is the midpoint of side $$BC$$, $$\overrightarrow{BA}=\overrightarrow{a}$$, $$\overrightarrow{BC}=\overrightarrow{b}$$, then $$\overrightarrow{DA}=$$___.

\overrightarrow{a}-\dfrac{1}{2}\overrightarrow{b}

math_2842

<image> As shown in the figure, it is a flowchart of an algorithm. When the input value of $$x$$ is $$5$$, the output result is ___.

2

math_6980

<image> Eight small cubes of the same size are glued together to form a larger cube as shown in Figure 1. The three views of the resulting geometric shape are shown in Figure 2. If Xiao Ming removes several of the eight small cubes, keeping the remaining small cubes in their original positions, and ensures that the three views of the new geometric shape are still as shown in Figure 2, then the maximum number of small cubes he can remove is ______.

4

math_6235

<image> In the Cartesian coordinate system, the vertices of the square $ABCD$ have coordinates $A$(1, 1), $B$(1, -1), $C$(-1, -1), $D$(-1, 1), and there is a point $P$(0, 2) on the $y$-axis. Construct the point $P_1$ symmetric to point $P$ with respect to point $A$, the point $P_2$ symmetric to point $P_1$ with respect to point $B$, the point $P_3$ symmetric to point $P_2$ with respect to point $C$, the point $P_4$ symmetric to point $P_3$ with respect to point $D$, the point $P_5$ symmetric to point $P_4$ with respect to point $A$, and the point $P_6$ symmetric to point $P_5$ with respect to point $B$, and so on. What are the coordinates of $P_{2020}$?

(0, 2)

math_1375

<image> The oblique axonometric drawing of a horizontally placed triangle is an isosceles right triangle A′B′O′. If O′B′ = 1, then what is the area of the original △ABO?

\sqrt{2}

math_2898

<image> As shown in the figure, the perimeter of rhombus ABCD is 40 cm, and the length of the diagonal AC is 10 cm. What are the measures of the adjacent interior angles of the rhombus in degrees?

60, 120

math_6701

<image> As shown in the figure, a right-angled triangle with a $$45^{\circ}$$ angle has two vertices placed on the opposite edges of a ruler. If $$\angle 1=20^{\circ}$$, then the degree measure of $$\angle 2$$ is ___ degrees.

25

math_5382

<image> As shown in the figure, in △ABC, BC = 5 cm; △ABC is translated along the line containing BC to △DEF (see figure); if EC = 2 cm, then CF = cm.

3

math_1153

<image> As shown in the figure, Uncle Wu has a piece of land in the shape of an equilateral triangle ABC. It is known that E and F are the midpoints of sides AB and AC, respectively, and EF measures 5 meters. He wants to enclose the quadrilateral BCFE with a fence to raise chicks. What is the length of the fence needed in meters?

25

math_6091

<image> As shown in the figure, Xiao Ming stands at point $$C$$ to observe points $$A$$ and $$E$$, the top points of buildings Jia and Yi, respectively. Points $$C$$, $$E$$, and $$A$$ are collinear, and points $$B$$ and $$D$$ are directly below points $$E$$ and $$A$$, respectively, with points $$D$$, $$B$$, and $$C$$ also collinear. If the distance between point $$B$$ and point $$C$$ is $$\quantity{20}{m}$$, the distance between point $$D$$ and point $$C$$ is $$\quantity{40}{m}$$, and the height of building Yi $$BE$$ is $$\quantity{15}{m}$$, then the height of building Jia $$AD$$ is ___$$\unit{m}$$ (Xiao Ming's height is negligible).

30

math_1198

<image> Execute the following pseudocode, the output value of $$x$$ is ___.

25

math_1373

<image> The figure below shows the frequency distribution histogram of the speeds of cars passing through a certain speed monitoring point on National Highway 204. Among the 300 cars passing through this monitoring point, approximately how many cars have speeds in the range [60, 80)?

150

math_6987

<image> In response to the 'Red Songs Resound in China' campaign, a town held a 'Red Song' singing competition. The organizing committee stipulates: any participant's score $$x$$ must satisfy: $$60 \leqslant x < 100$$. After the competition, the scores of all participants were organized as shown in Table (1). According to the information provided in Table (1), $$n=$$______.

0.3

math_1990

<image> The graph of the function $$y=f(x)$$ is shown below. Which of the following statements is correct? 1. $$f'(3) > 0$$; 2. $$f'(3) < 0$$; 3. $$f'(3)=0$$; 4. The sign of $$f'(3)$$ is uncertain.

2

math_5320

<image> As shown in the figure, the graph of the function $$f(x)$$ is the line segment $$ABC$$, where the coordinates of points $$A$$, $$B$$, and $$C$$ are $$(0,4)$$, $$(2,0)$$, and $$(6,4)$$, respectively. Then $$f(f(f(2)))=$$ ___.

2

math_5786

<image> As shown in the figure, if the parabola $y=ax^2+bx+c$ has no intersection points with the $x$-axis, then $a, b, c$ should satisfy the relationship:

b^2-4ac < 0

math_6394

<image> After the computer executes the following pseudocode, if the input is $$a=2$$, $$b=3$$, what is the output?

6

math_2017

<image> As shown in the figure, after running the pseudocode of the algorithm, the value of $$S$$ that is output is ___.

17

math_4135

<image> The figure below has ( ______ ) lines of symmetry.

1

math_7282

<image> In the 'Chinese Classics Recitation' activity, a school conducted a random survey of 50 students to understand the extracurricular reading situation of the entire school's 1300 students. The survey collected data on the students' weekly extracurricular reading time and created the bar chart shown in the figure. Based on the data in the chart, estimate the number of students among the 1300 who read for at least 7 hours per week.

520

math_4417

<image> As shown in the figure, the curve $$C$$: $$y=\sqrt{x}$$, suppose the line $$l_{1}$$ is tangent to the curve $$C$$ at point $$P$$, and the line $$l_{2}$$ passes through point $$P$$ and is perpendicular to line $$l_{1}$$. If the line $$l_{2}$$ intersects the $$x$$-axis at point $$Q$$, and the projection of point $$P$$ on the $$x$$-axis is point $$R$$, then $$\left \lvert RQ\right \rvert =$$___.

\dfrac{1}{2}

math_6052

<image> As shown in the figure, in △ABC, ∠ACB = 90°, ∠B = 30°, CD is perpendicular to AB at point D, AB = 4, then the length of BD is.

3

math_6815

<image> The following is a statistical chart based on the donation situation of a middle school for earthquake-stricken areas. It is known that the school has 3000 students. Calculate the total amount of donations made by the school.

37770

math_5505

<image> Xiaoliang conducted a questionnaire survey on water-saving methods among 60 classmates (each choosing one method). The number of people is shown in the chart. If this data is represented in a pie chart, what is the central angle of the sector representing 'Multiple Uses of Water' in degrees?

240

math_4743

<image> Given that the maximum speed limit on a certain road is $$\quantity{50}{km/h}$$, the electronic monitoring system measures the speeds of 6 consecutive cars as shown in the figure. If 2 cars are randomly selected, the probability that exactly 1 car is speeding is ___.

\dfrac{8}{15}

math_7878

<image> There are four game boards. If a soybean falls on the shaded area, a prize can be won. Xiao Ming hopes to win a prize. He should choose the game board __ (fill in the number).

1

math_1421

<image> As shown in the figure, the function $$f(x)=x^{n^{2}-2n-3}(n \in \mathbf{Z})$$ is illustrated. What is the value of $$n$$?

1

math_5478

<image> As shown in the figure, $$AB\parallel CD$$, $$PB$$ bisects $$\angle ABC$$, $$PC$$ bisects $$\angle DCB$$. Then $$\angle P=$$ ___ degrees.

90

math_2

<image> As shown in the figure, △ABC is inscribed in circle ⊙O, BD is the diameter of ⊙O, ∠CBD = 21°, then the measure of ∠A is ___ degrees?

69

math_3551

<image> Given the graph of the linear function $y=kx+b$ as shown, the range of $y$ when $x < 0$ is.

y < -2

math_2467

<image> As shown in the figure, in rectangle ABCD, BC=4, AB=$2\sqrt{3}$, and P is a point on the extension of AD. ${{S}_{\Delta PBC}}=$ ___?

4 \sqrt{3}

math_3835

<image> Teacher Li copied the probability distribution of a random variable $$\xi$$ from a textbook as shown in the table: Student Wang was asked to calculate the mathematical expectation of $$\xi$$. Although the value at the "?" is completely illegible and the two "!"s are blurry, it can be determined that the two "!"s have the same value. Based on this, Student Wang correctly gave the answer $$E \xi =$$___.

2

math_6585

<image> On April 20, 2013, a major earthquake of magnitude 7.0 struck Lushan County in Ya'an City. The people of the entire Sichuan province united to respond to the disaster. A class organized a 'Donate Pocket Money, Show Love' activity. The donation situation of the 50 students in the class is shown in the figure below. What is the mode of the donation amounts?

20

math_707

<image> In the right triangle △ABC, ∠BAC=90°. Triangle ABC is rotated clockwise around point A to form triangle ADE, with point B's corresponding point D landing exactly on BC. If AC=4$\sqrt{3}$ and ∠B=60°, then the length of CD is

4

math_3085

<image> The graph of an inverse proportion function in the third quadrant is shown in the figure. A is any point on the graph, AM is perpendicular to the x-axis at point M, and O is the origin. If the area of △AOM is 3, then the expression of this inverse proportion function is ______.

y = 6/x

math_234

<image> If the side length of a regular hexagon is 3, then the length of its longer diagonal is.

6

math_4091

<image> In the figure, point O is a point inside △ABC, and the distances from point O to the three sides AB, BC, and CA are equal. Connecting OB and OC, if ∠A = 78°, then the degree measure of ∠BOC is ___ degrees.

129

math_1250

<image> Lelé found that three balls of the same size can fit perfectly inside a cylindrical box (the thickness of the bottom and lid is negligible), as shown in the figure. What fraction of the total volume of the box is occupied by the three balls? (The volume of a sphere is given by the formula $V=\frac{4}{3}\pi r^3$)

\frac{2}{3}

math_3157

<image> A research team obtained a set of data in an experiment and organized it into the scatter plot shown below. Among the following functions, which one best approximates the relationship between y and t? (Fill in the sequence number). 1. y = 2^t; 2. y = 2t^2; 3. y = t^3; 4. y = log_2 t.

4

math_5512

<image> The perspective view of a geometric solid is shown in the figure. Among the following four top views, which one is correct?

2

math_90

<image> In the figure, in rectangle ABCD, BE is perpendicular to AC, intersecting AC and AD at points F and E, respectively. Given AF = 2 and AC = 6, what is the length of AB?

2 \sqrt{3}

math_3803

<image> In the figure, $$\triangle ABC$$, $$DE$$ is the perpendicular bisector of $$AC$$, $$AE = \quantity{6}{cm}$$, and the perimeter of $$\triangle ABD$$ is $$\quantity{26}{cm}$$, then the perimeter of $$\triangle ABC$$ is ___$$\unit{cm}$$.

38

math_2884

<image> As shown in the figure, in $\vartriangle ABC$, $\angle B = \angle C = 60{}^\circ$, point $D$ is on side $AB$, $DE \perp AB$, and intersects side $AC$ at point $E$. If $AD = 1$ and $BC = 6$, then $CE =$．

4

math_241

<image> As shown in the figure, in the pentahedron $$ABCDEF$$, $$FA \perp$$ plane $$ABCD$$, $$AD \parallel BC \parallel FE$$, $$AB \perp AD$$, $$AF=AB=BC=FE=\dfrac{1}{2}AD$$. What is the angle between the skew lines $$BF$$ and $$ED$$ in degrees?

60

math_4005

<image> As shown in the figure, if the input number is "$$37$$", then the output result is ___.

73

math_4146

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