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758c47e7-4fc2-47a8-829a-ad83175868ae
|
<image>As shown in the figure, the medians $$AD$$, $$BE$$, and $$CF$$ of $$\triangle ABC$$ intersect at point $$G$$. If $$S_{\triangle ABC} = 12$$, then the area of the shaded region is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/758c47e7-4fc2-47a8-829a-ad83175868ae.png"
}
] |
c09ec00e-a7bb-4861-91e6-db317c4d8be9
|
<image>Given that the graph of the function $$f(x)$$ is the broken line segment $$OAB$$ as shown in the figure below, with point $$A$$ having coordinates $$(1,2)$$ and point $$B$$ having coordinates $$(3,0)$$, define the function $$g(x)=f(x)\cdot (x-1)$$. Then the expression for the function $$g(x)$$ is ___.
|
$$g(x)=\begin{cases} 2x^{2}-2x, & 0 \leqslant x < 1, \\ -x^{2}+4x-3, & 1 \leqslant x \leqslant 3 \end{cases}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c09ec00e-a7bb-4861-91e6-db317c4d8be9.png"
}
] |
8efaae13-ca51-41bb-8e57-dbc0db50542e
|
<image>The tour route of a scenic area is shown in the figure. A person enters from point $$P$$ and exits from point $$Q$$, visiting three attractions $$A$$, $$B$$, and $$C$$ along the route. The number of different tour routes without repetition (except for intersection point $$O$$) is ___ kinds.
|
$$48$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8efaae13-ca51-41bb-8e57-dbc0db50542e.png"
}
] |
d2cd7194-a024-43d9-9771-fe6a7214977b
|
<image>As shown in the figure, in $$\triangle ABC$$, $$AD$$ is the altitude on side $$BC$$, $$\angle C=45^{\circ}$$, $$\sin B=\dfrac{1}{3}$$, and $$AD=1$$. Find the length of $$BC$$.
|
$$2\sqrt{2}+1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d2cd7194-a024-43d9-9771-fe6a7214977b.png"
}
] |
95d2a454-6dfe-4686-bca0-efd241483990
|
<image>As shown in the figure, in isosceles triangle $$ABC$$, $$AB=AC$$, $$DE$$ is the perpendicular bisector of $$AB$$. Given that $$\angle ADE=40^{\circ}$$, then $$\angle DBC=$$___$$^{\circ}$$.
|
$$15$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/95d2a454-6dfe-4686-bca0-efd241483990.png"
}
] |
ec857ee1-07d6-4457-b655-2a43ddb921c6
|
<image>In the right trapezoid $$ABCD$$, $$AD \perp DC$$, $$AD \parallel BC$$, $$BC=2CD=2AD=2$$. If the right trapezoid is rotated around the side $$BC$$ for one complete revolution, then the surface area of the resulting solid is ___.
|
$$(3+\sqrt{2}) \pi$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ec857ee1-07d6-4457-b655-2a43ddb921c6.png"
}
] |
c95e1b22-d7ed-43ab-84fb-9690757f5455
|
<image>As shown in the figure, in $$\triangle ABC$$, the angle bisectors of $$\angle B$$ and $$\angle C$$ intersect at point $$O$$. A line $$DE \parallel BC$$ is drawn through point $$O$$, intersecting sides $$AB$$ and $$AC$$ at points $$D$$ and $$E$$, respectively. If $$AB=5$$ and $$AC=4$$, then the perimeter of $$\triangle ADE$$ is ___.
|
$$9$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c95e1b22-d7ed-43ab-84fb-9690757f5455.png"
}
] |
18d6607e-73df-4614-86eb-89f2e8af9b71
|
<image>As shown in the figure, a circle with radius $$r$$ and a sector with radius $$R$$ are cut from a square piece of paper, and they are precisely used to form the cone shown in the figure. The relationship between $$R$$ and $$r$$ is ______.
|
$$R=4r$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/18d6607e-73df-4614-86eb-89f2e8af9b71.png"
}
] |
23fba490-1d7e-4b14-bdb2-4141c53f5951
|
<image>As shown in the figure, in $$\triangle ABC$$, $$AB=AC$$, $$BC=\quantity{12}{cm}$$, point $$D$$ is on $$AC$$, $$DC=\quantity{4}{cm}$$, and line segment $$DC$$ is translated $$\quantity{7}{cm}$$ in the direction of $$CB$$ to obtain line segment $$EF$$, with points $$E$$ and $$F$$ landing on sides $$AB$$ and $$BC$$ respectively. What is the perimeter of $$\triangle EBF$$?
|
$$13$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/23fba490-1d7e-4b14-bdb2-4141c53f5951.png"
}
] |
757a85d8-c91f-41b0-8d6c-6aa726c50d03
|
<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 $$\angle BAC=40\unit{^{ \circ }}$$, the distance from point $$A$$ to $$BC$$ is ___ $$\unit{cm}$$ (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).
|
$$\number{14.1}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/757a85d8-c91f-41b0-8d6c-6aa726c50d03.png"
}
] |
55868cc9-4ea6-4684-aebe-84d469fa1262
|
<image>As shown in the figure, in quadrilateral $$ABCD$$, $$AD \parallel BC$$. If $$\angle A=110^{ \circ }$$, then $$\angle B=$$ ___.
|
$$70^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/55868cc9-4ea6-4684-aebe-84d469fa1262.png"
}
] |
03ead4ec-71a0-4624-ab86-a77db941ce21
|
<image>The domain of the function $$f(x)$$ is the open interval $$(a,b)$$, and the graph of the derivative function $$f'(x)$$ within $$(a,b)$$ is shown in the figure. How many minimum points does the function $$f(x)$$ have in the open interval $$(a,b)$$?
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/03ead4ec-71a0-4624-ab86-a77db941ce21.png"
}
] |
edd4ec67-4bcf-4724-851d-c04d62ae53ce
|
<image>The figure is a plane shape composed of rays $$AB$$, $$BC$$, $$CD$$, $$DE$$, and $$EA$$. Then $$ \angle 1+ \angle 2+ \angle 3+ \angle 4+ \angle 5=$$ ___.
|
$$360^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/edd4ec67-4bcf-4724-851d-c04d62ae53ce.png"
}
] |
1fa21bf4-06a1-42dd-a032-1126f5d3b639
|
<image>As shown in the figure, the regular hexagon $$ABCDEF$$ is inscribed in $$\odot{}O$$, with the radius of $$\odot{}O$$ being $$1$$. The length of the arc $$\overset{\frown} {AB}$$ is ___.
|
$$\dfrac{\pi }{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1fa21bf4-06a1-42dd-a032-1126f5d3b639.png"
}
] |
e33b2538-820c-4484-ae03-6c7bb6046fe9
|
<image>As shown in Figure 1, in the rectangular paper piece $$ABCD$$, $$AB=5$$, $$BC=3$$. First, perform the operation as shown in Figure 2: fold the rectangular paper piece $$ABCD$$ along a line passing through point $$A$$ so that point $$D$$ falls on point $$E$$ on side $$AB$$, with the crease being $$AF$$; then, perform the operation as shown in Figure 3: fold along a line passing through point $$F$$ so that point $$C$$ falls on point $$H$$ on $$EF$$, with the crease being $$FG$$. The distance between points $$A$$ and $$H$$ is ___.
|
$$\sqrt{10}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e33b2538-820c-4484-ae03-6c7bb6046fe9.png"
}
] |
f9310deb-a597-45ff-8985-931b6c706e94
|
<image>The intuitive diagram of a certain triangle is an isosceles right triangle with a hypotenuse of $$2$$, as shown in the figure. Then the area of the original triangle is ___.
|
$$2\sqrt{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f9310deb-a597-45ff-8985-931b6c706e94.png"
}
] |
c5faa072-2251-4532-b073-5d74ce390842
|
<image>As shown in the figure, $\Delta AOC$ and $\Delta AOB$ are symmetric with respect to line $AO$, and $\Delta DOB$ and $\Delta AOB$ are symmetric with respect to line $BO$. $OC$ intersects $BD$ at point $E$. If $\angle C=15{}^\circ $ and $\angle D=25{}^\circ $, then the measure of $\angle BEC$ is.
|
95°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c5faa072-2251-4532-b073-5d74ce390842.png"
}
] |
c54b9937-3496-48bf-9d8e-9b8b12565290
|
<image>In the rectangle ABCD, points E and F are on sides AD and DC, respectively, with BE⊥EF. Given AB=6, AE=9, and DE=2, find the length of EF.
|
$\sqrt{13}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c54b9937-3496-48bf-9d8e-9b8b12565290.png"
}
] |
a16517f2-503f-49f9-9f5e-80f19b02bb6a
|
<image>As shown in the figure, points A, O, and B are on a straight line, and ∠AOD = 35°, OD bisects ∠AOC. Therefore, ∠BOC = ___ degrees.
|
110
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a16517f2-503f-49f9-9f5e-80f19b02bb6a.png"
}
] |
f4489d00-b860-4add-a9e4-3e6d291c4ff5
|
<image>The figure shows the net of a cube, with the faces labeled with the numbers 1, 2, 3, -3, A, and B. The numbers on opposite faces are additive inverses of each other. What is B?
|
-2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f4489d00-b860-4add-a9e4-3e6d291c4ff5.png"
}
] |
4ced1430-82c1-4a52-a3e2-37f5f89ff3e7
|
<image>Let the universal set be $U$, and use the symbols for the intersection, union, and complement of sets $A$ and $B$ to represent the shaded area in the figure .
|
$(A\bigcup B)\bigcap {{C}_{U}}(A\bigcap B)$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4ced1430-82c1-4a52-a3e2-37f5f89ff3e7.png"
}
] |
5a16e299-fb94-48bb-a0fc-8c6847ee7597
|
<image>A school's science group went on a field trip and encountered a wetland several meters wide. To safely and quickly cross the wetland, they laid several wooden boards along their path, forming a temporary passage. The pressure $p(\text{Pa})$ exerted by the wooden boards on the ground is an inverse proportion function of the area $S\left( {{m}^{2}} \right)$ of the boards, as shown in the graph. When the pressure exerted by the wooden boards on the ground does not exceed 6000$\text{Pa}$, the area of the wooden boards should be at least.
|
$0.1{{m}^{2}}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5a16e299-fb94-48bb-a0fc-8c6847ee7597.png"
}
] |
08f54a98-a7af-4084-917a-7d7da97be49c
|
<image>The figure below shows the stacked bar chart obtained from a survey on whether male and female students in the first year of a high school like hiking. The shaded part represents the frequency of students who like hiking. It is known that there are 500 male students and 400 female students in the grade (assuming all students participated in the survey). Now, 23 students are to be selected from all the students who like hiking using stratified sampling. The number of male students to be selected is:
|
15
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/08f54a98-a7af-4084-917a-7d7da97be49c.png"
}
] |
5ee79c47-cbd0-4ed7-b4eb-ee64ef91d8ab
|
<image>In the steel frame shown in the figure, ∠A = $x$ degrees, and equal-length steel bars ${{P}_{1}}{{P}_{2}}, {{P}_{2}}{{P}_{3}}, {{P}_{3}}{{P}_{4}}, {{P}_{4}}{{P}_{5}}$... are welded to reinforce the frame. If ${{P}_{1}}A = {{P}_{1}}{{P}_{2}}$, and at most 6 such steel bars are required, then the range of values for $x$ is.
|
$\frac{90}{7}\le x < 15$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5ee79c47-cbd0-4ed7-b4eb-ee64ef91d8ab.png"
}
] |
30dfdeb3-dc95-44cf-af2a-5d5b8236d7b4
|
<image>As shown in the figure, the oblique dimetric perspective drawing of the horizontally placed △ABC is △A'B'C' in the figure. Given that A'C' = 6 and B'C' = 4, the actual length of side AB is.
|
10
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/30dfdeb3-dc95-44cf-af2a-5d5b8236d7b4.png"
}
] |
44fcab1b-9d0e-4e21-9512-0e5f1bb40516
|
<image>In the figure, in $\vartriangle ABC$, $D$ is a point on $AC$, and $\frac{CD}{AD}=\frac{1}{2}$. A line through point $D$ parallel to $BC$ intersects $AB$ at point $E$. Line segment $CE$ is drawn, and a line through point $D$ parallel to $CE$ intersects $AB$ at point $F$. If $AB=15$, then $EF=$.
|
$\frac{10}{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/44fcab1b-9d0e-4e21-9512-0e5f1bb40516.png"
}
] |
f7e51a03-e1d7-4731-b231-085f49b2d4fb
|
<image>Xiao Ming hangs the rectangular board ABCD on the wall as shown in the figure, with E being the midpoint of AD, and ∠ABD = 60°. He uses it to play a dart game (each dart always lands on the board). The probability of hitting the shaded area is.
|
$\frac{1}{8}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f7e51a03-e1d7-4731-b231-085f49b2d4fb.png"
}
] |
eee4e0ef-3c60-4b83-a9bf-ac37a7d46ab1
|
<image>As shown in the figure, construct ∠EAF = 45° inside the square ABCD. AE intersects BC at point E, and AF intersects CD at point F. Connect EF, and draw AH perpendicular to EF, with H as the foot of the perpendicular. Rotate △ADF 90° clockwise around point A to get △ABG. If BE = 2 and DF = 3, then the length of AH is.
|
6
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/eee4e0ef-3c60-4b83-a9bf-ac37a7d46ab1.png"
}
] |
b42fbd0e-8ba7-4e61-9773-e3406c6bfbba
|
<image>Execute the program flowchart shown in the figure. If the output is $$S=88$$, then the condition that should be filled in the judgment box is ___.
|
$$k>5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b42fbd0e-8ba7-4e61-9773-e3406c6bfbba.png"
}
] |
da1c1eb9-59a1-4f92-b997-85023250def8
|
<image>As shown in the figure, lines $$AB$$ and $$CD$$ intersect at point $$O$$, $$OE \perp CD$$, and $$O$$ is the foot of the perpendicular. If $$\angle AOC = 62\degree$$, then $$\angle BOE =$$ ______ degrees.
|
$$28$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/da1c1eb9-59a1-4f92-b997-85023250def8.png"
}
] |
698c2058-12fb-42e6-ba7d-c4f3388d38af
|
<image>The shooting scores (unit: points) of two athletes, Athlete A and Athlete B, from 5 training sessions are as follows: Which athlete has more stable scores (smaller variance)? The variance of the more stable athlete's scores is ______.
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/698c2058-12fb-42e6-ba7d-c4f3388d38af.png"
}
] |
eec8c8fc-5f5b-4d0a-af1b-26e97ddb6769
|
<image>As shown in the figure, the area of $$\triangle ABC$$ is $$36$$. Translate $$\triangle ABC$$ along $$BC$$ to $$\triangle A'B'C'$$ so that point $$B'$$ coincides with point $$C$$. Connect $$AC'$$, which intersects $$AC$$ at point $$D$$. The area of $$\triangle C'DC$$ is ___.
|
$$18$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/eec8c8fc-5f5b-4d0a-af1b-26e97ddb6769.png"
}
] |
c0bb7ad6-bd54-4f67-82fc-0b5c4451cb25
|
<image>As shown in the figure, in the parallelepiped $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$E$$ is the center of the square base $$ABCD$$. If $$\overrightarrow{{A}_{1}E}=\overrightarrow{{A}_{1}A}+x\overrightarrow{{A}_{1}B_{1}}+y\overrightarrow{{A}_{1}D_{1}}$$, then $$x+y=$$ ___.
|
$$1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c0bb7ad6-bd54-4f67-82fc-0b5c4451cb25.png"
}
] |
e5d2e37b-8e07-42cb-adc0-182c535fa49f
|
<image>As shown in the figure, $$\angle AOB$$ is placed on a square grid. The value of $$\cos \angle AOB$$ is ___.
|
$$\dfrac{\sqrt{5}}{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e5d2e37b-8e07-42cb-adc0-182c535fa49f.png"
}
] |
de9aa55e-0e48-419b-a332-f9cf22f587ef
|
<image>As shown in the figure, Xiao Ming starts from point $$E$$ on the street, first meets Xiao Hong at point $$F$$, and then they go together to the nursing home located at point $$G$$ to participate in volunteer activities. The number of shortest paths Xiao Ming can choose to reach the nursing home is ___.
|
$$18$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/de9aa55e-0e48-419b-a332-f9cf22f587ef.png"
}
] |
4f987c7f-6408-4a85-aacf-93eff0ad9f06
|
<image>As shown in the figure, in the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$AB=2D_{1}D=2$$, $$DA<1$$, and $$E$$ is a moving point on line segment $$AB$$. When $$DE=1$$ and $$DE⊥C_{1}E$$, the angle formed between $$DE$$ and $$C_{1}D_{1}$$ is ___.
|
$$60^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4f987c7f-6408-4a85-aacf-93eff0ad9f06.png"
}
] |
eff84e2e-302e-4f09-895c-1b50d2d53c59
|
<image>Given that the perimeter of $$\triangle ABC$$ is $$50\ \unit{cm}$$, the midsegment $$DE=8\ \unit{cm}$$, and the midsegment $$DF=10\ \unit{cm}$$, then the length of the other midsegment $$EF$$ is ___$$\unit{cm}$$.
|
$$7$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/eff84e2e-302e-4f09-895c-1b50d2d53c59.png"
}
] |
c938edfc-346f-4bed-8db8-6122157d3df8
|
<image>As shown in the figure, in the spatial quadrilateral $ABCD$, $AC$ and $BD$ are diagonals, $G$ is the centroid of $\Delta ABC$, and $E$ is a point on $BD$ such that $BE=3ED$. Using $|\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}|$ as the basis, then $\overrightarrow{GE}=$.
|
$-\frac{1}{12}\overrightarrow{AB}-\frac{1}{3}\overrightarrow{AC}+\frac{3}{4}\overrightarrow{AD}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c938edfc-346f-4bed-8db8-6122157d3df8.png"
}
] |
69360d6e-f179-4c42-a3e0-81341a7b9a66
|
<image>A quality inspection was conducted on a batch of products (unit: grams), with a sample size of 1600. The frequency distribution histogram of the inspection results is shown in the figure. According to the standard, products with a single piece quality in the range [25, 30) are first-class products, those in the ranges [15, 20), [20, 25), and [30, 35) are second-class products, and the rest are third-class products. The number of third-class products in the sample is:
|
200
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/69360d6e-f179-4c42-a3e0-81341a7b9a66.png"
}
] |
913216a1-d312-4b5e-a58e-410223dff196
|
<image>As shown in the figure, the circuit has three switches $a$, $b$, and $c$. The probability of each switch being on or off is $\frac{1}{2}$, and they are mutually independent. The probability that bulb 甲 (Jia) lights up is:
|
$\frac{1}{8}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/913216a1-d312-4b5e-a58e-410223dff196.png"
}
] |
9d1abe41-cd62-4cb8-96a7-d63d9233034d
|
<image>As shown in the figure, the tangent line to the graph of the function $f(x)$ at point $P$ is: $y=-2x+5$. Then $f(2) + f'(2)=$.
|
-1
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9d1abe41-cd62-4cb8-96a7-d63d9233034d.png"
}
] |
1567c312-66b3-4fe9-9c9d-ac1fc5318887
|
<image>Execute the program flowchart as shown. If the input is $S=1$, $k=1$, then the output $S=$．
|
57
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1567c312-66b3-4fe9-9c9d-ac1fc5318887.png"
}
] |
52836b1b-044d-4150-9df7-b04e99d56396
|
<image>Given, as shown in the figure, three lines AB, CD, and EF intersect at point O, and $CD \bot EF$, $\angle AOC = 20{}^\circ$. If OG bisects $\angle BOF$, then $\angle DOG =$.
|
$55{}^\circ$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/52836b1b-044d-4150-9df7-b04e99d56396.png"
}
] |
b2ad304d-6817-4325-8b93-9670bf4e9381
|
<image>As shown in the figure, the graph of the function $f(x)$ consists of two line segments, with its domain being $[-1,0) \cup (0,1]$. The set of values of $x$ that satisfy the inequality $|f(x) - f(-x)| \ge 1$ is.
|
$\left[ -\frac{1}{2},0 \right) \cup \left( 0,\frac{1}{2} \right]$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b2ad304d-6817-4325-8b93-9670bf4e9381.png"
}
] |
160928e9-a983-470f-b3f3-096278230f5d
|
<image>In $\vartriangle ABC$, AD is the altitude on side BC, $BC=12$, $AD=8$. The vertices E and F of square EFGH lie on AB and AC, respectively, while H and G lie on BC. What is the side length of square EFGH?
|
4.8
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/160928e9-a983-470f-b3f3-096278230f5d.png"
}
] |
58411882-f1f7-48b1-a2d5-fcae70afd423
|
<image>As shown in the figure, in the plane quadrilateral $ABCD$, $AB=\sqrt{2}$, $BC=\sqrt{3}$, $AB\bot AD$, $AC\bot CD$, $AD=3AC$, then $AC=$.
|
3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/58411882-f1f7-48b1-a2d5-fcae70afd423.png"
}
] |
d5f4c62d-d3db-4140-8a4e-cf39a906dc89
|
<image>As shown in the figure, if set $A=\left\{ 1,2,3,4,5 \right\}$, $B=\left\{ 2,4,6,8,10 \right\}$, then the set represented by the shaded area in the figure (expressed using the listing method) is:
|
$\left\{ 6,8,10 \right\}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d5f4c62d-d3db-4140-8a4e-cf39a906dc89.png"
}
] |
3860bbad-0b14-4228-b556-48985ded553d
|
<image>As shown in the figure, $\vartriangle ABC$ is an equilateral triangle, and $P$ is any point inside $\vartriangle ABC$. $PD\parallel AB$, $PE\parallel BC$, $PF\parallel AC$. If the perimeter of $\vartriangle ABC$ is $12cm$, then $PD+PE+PF=$$cm$.
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3860bbad-0b14-4228-b556-48985ded553d.png"
}
] |
916aaafb-3766-4cd9-ac67-8811a45c2341
|
<image>As shown in the figure, AB is the diameter of circle O, and CD is a chord of circle O, ∠DCB = 32°. Then ∠ABD =
|
58°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/916aaafb-3766-4cd9-ac67-8811a45c2341.png"
}
] |
f04d498d-9cad-44fc-9c18-7686a635d2af
|
<image>Given the functions $f(x) = 2^{x+1}$ and $g(x) = 2^x$ with their graphs as shown, the line $y = a$ intersects the graphs of the two functions at points A and B, respectively. If there exists a point C on the function $y = g(x)$ such that it forms an equilateral triangle $\Delta ABC$, then $a =$.
|
$\frac{2\sqrt{3}+\sqrt{6}}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f04d498d-9cad-44fc-9c18-7686a635d2af.png"
}
] |
e924fcdb-e4d2-4bac-8597-12480f9a9c2f
|
<image>If the amount paid for a taxi ride $y$ (yuan) is a function of the distance traveled $x$ (kilometers), and the function graph is composed of line segment $AB$, line segment $BC$, and ray $CD$ (as shown in the figure), then the amount to be paid for a ride of 8 kilometers is ______ yuan.
|
26
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e924fcdb-e4d2-4bac-8597-12480f9a9c2f.png"
}
] |
bf3cd628-4470-4cef-a01e-2db76723673b
|
<image>As shown in the figure, in the regular hexagon ABCDEF, line segments AD and BE intersect at point G. Circles O$_{1}$ and O$_{2}$ are the incircles of triangles ABG and DEG, respectively. Circles O$_{3}$ and O$_{4}$ are the incircles of quadrilaterals BCDG and AGEF, respectively. If a point is randomly thrown into the hexagon ABCDEF, the probability that the point lands in the shaded area is.
|
$\frac{13\sqrt{3}\pi }{108}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bf3cd628-4470-4cef-a01e-2db76723673b.png"
}
] |
091bf953-ab43-4ba4-8e61-9f71ba4b266f
|
<image>The flowchart (i.e., algorithm flowchart) is shown in the figure. Its output result is.
|
129
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/091bf953-ab43-4ba4-8e61-9f71ba4b266f.png"
}
] |
2fbc1035-a0a5-4b40-b627-29d79175a139
|
<image>As shown in the figure, ∠C=90°. The right triangle ABC is translated 5 cm along the ray BC to get △A'B'C'. Given BC=3 cm and AC=4 cm, the area of the shaded region is cm².
|
14
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2fbc1035-a0a5-4b40-b627-29d79175a139.png"
}
] |
cd37bb1f-b4f9-4810-8de2-7b00af0d764e
|
<image>As shown in the figure, the height of a utility pole AB is 10 meters. When the angle between the sunlight and the ground is 60°, the length of its shadow AC is meters.
|
$\frac{10\sqrt{3}}{3}\cdot $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cd37bb1f-b4f9-4810-8de2-7b00af0d764e.png"
}
] |
8a6592d2-2e85-4304-89bc-bb0f3202cffc
|
<image>As shown in the figure, in the equilateral triangle $\vartriangle ABC$, point $D$ is a point on side $BC$, and $DE \bot BC$ intersects $AB$ at $E$, $DF \bot AC$ intersects $AC$ at $F$. The measure of $\angle EDF$ is.
|
60°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8a6592d2-2e85-4304-89bc-bb0f3202cffc.png"
}
] |
37c0c940-8269-454e-bbab-3ff2eda84dd3
|
<image>In the circle ⊙O, the diameter EF is perpendicular to CD, with the foot of the perpendicular being M. If CD = 2 and EM = 4, then the radius of ⊙O is.
|
$\frac{17}{8}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/37c0c940-8269-454e-bbab-3ff2eda84dd3.png"
}
] |
1ce4cd20-185f-4b78-9556-3a587563f119
|
<image>The three views of a composite solid are shown in the figure. Each grid represents a unit square. The surface area of this geometric solid is:
|
$45\pi $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1ce4cd20-185f-4b78-9556-3a587563f119.png"
}
] |
7b3b516f-cb6a-43f1-bf93-66b25890aaf6
|
<image>The domain of the function $y=f(x)$ is $[-1,1]$, and its graph is shown in the figure. If the inverse function of $y=f(x)$ is $y=f^{-1}(x)$, then the solution set of the inequality $(f(x)-\frac{1}{2})(f^{-1}(x)-\frac{1}{2}) > 0$ is
|
$(\frac{3}{4},1]$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7b3b516f-cb6a-43f1-bf93-66b25890aaf6.png"
}
] |
05e9704c-560b-4668-8dd0-a7b2af932b6a
|
<image>In plane $\alpha$, there is a circle $ACB$ (as shown in the figure), with $AB$ as the diameter, $SA \bot \alpha$, and $C$ is a point on the arc $\overset\frown{AB}$. If $AC:AB:SA=1:2:2$, then the cosine value of the dihedral angle $C-AB-A$ is.
|
$\frac{\sqrt{15}}{5}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/05e9704c-560b-4668-8dd0-a7b2af932b6a.png"
}
] |
9412362a-0f90-4c08-9dbf-1148745969ad
|
<image>As shown in the figure, the pattern is arranged according to a certain rule. According to this rule, in the patterns from the 1st to the 2012th, there are ______ '♣' symbols.
|
503
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9412362a-0f90-4c08-9dbf-1148745969ad.png"
}
] |
07f1a93e-da93-4bfe-91b2-c02fc8d47702
|
<image>As shown in the figure, the lines $$l_{1} \parallel l_{2} \parallel l_{3}$$, line $$AC$$ intersects $$l_{1}$$, $$l_{2}$$, $$l_{3}$$ at points $$A$$, $$B$$, $$C$$ respectively, and line $$DF$$ intersects $$l_{1}$$, $$l_{2}$$, $$l_{3}$$ at points $$D$$, $$E$$, $$F$$ respectively. $$AC$$ and $$DF$$ intersect at point $$G$$, and it is given that $$AG=2$$, $$GB=1$$, $$BC=5$$. What is the value of $$\dfrac{DE}{EF}$$?
|
$$\dfrac{3}{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/07f1a93e-da93-4bfe-91b2-c02fc8d47702.png"
}
] |
b16dc053-d7fb-4716-9f0b-938d01d2db4e
|
<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ABC = \angle ACB = 72^{\circ}$$, $$BD$$ and $$CE$$ are the angle bisectors of $$\angle ABC$$ and $$\angle ACB$$ respectively, and their intersection point is $$F$$. How many isosceles triangles are there in the figure?
|
$$8$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b16dc053-d7fb-4716-9f0b-938d01d2db4e.png"
}
] |
bc95c29a-29a8-4d1a-8b2f-0dcff9523466
|
<image>The Hundred Sons Return Chart is a square number table formed by the non-repeating arrangement of $$1, 2, 3, \cdots, 100$$. It is a numerical brief history of Macau, such as: the four central numbers "$$19 99 12 20$$" indicate the date of Macau's return, the middle two numbers "$$23 50$$" in the last row indicate the area of Macau, $$\cdots \cdots $$, and it is also a tenth-order magic square, where the sum of each row of $$10$$ numbers, each column of $$10$$ numbers, and each diagonal of $$10$$ numbers are all equal. What is this sum?
|
$$505$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bc95c29a-29a8-4d1a-8b2f-0dcff9523466.png"
}
] |
88186d7f-4dc9-4fe0-990b-fea520d47017
|
<image>As shown in the figure, in rectangle $$ABCD$$, $$E$$ is a point on $$BC$$, and $$DF \bot AE$$ at point $$F$$. If $$AB=4$$, $$AD=5$$, and $$AE=6$$, then the length of $$DF$$ is ___.
|
$$\dfrac{10}{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/88186d7f-4dc9-4fe0-990b-fea520d47017.png"
}
] |
16956e2a-0d6b-4f6f-9a2f-041d8f7db505
|
<image>As shown in the figure, the curves are the graphs of the logarithmic functions $$y=\log \nolimits_{a}x$$, $$y=\log \nolimits_{b}x$$, $$y=\log \nolimits_{c}x$$, and $$y=\log \nolimits_{d}x$$. The relationship between $$a$$, $$b$$, $$c$$, $$d$$, and $$1$$ is ___.
|
$$b>a>1>d>c$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/16956e2a-0d6b-4f6f-9a2f-041d8f7db505.png"
}
] |
f877d486-dbdb-445d-8566-3801085fe968
|
<image>As shown in the figure, lines $$AB$$ and $$CD$$ intersect at point $$O$$, and $$OM \perp AB$$. If $$\angle COB=135^{ \circ }$$, then $$\angle MOD=$$ ___ degrees.
|
$$45$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f877d486-dbdb-445d-8566-3801085fe968.png"
}
] |
72f3c25a-3171-47f2-9544-65892d4889fd
|
<image>As shown in the figure, in the isosceles trapezoid $$ABCD$$, $$AD \parallel BC$$, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. If $$OB=3$$, then $$OC=$$ ___.
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/72f3c25a-3171-47f2-9544-65892d4889fd.png"
}
] |
98ff0ce1-417e-4208-80b0-46951446b3ec
|
<image>Figure 1 illustrates an ancient Chinese mathematical problem about 'vine wrapping a tree, find the vine length.' The ancient method to solve this problem is: the length of the diagonal of the unfolded side of the geometric figure is the length of the vine. Try to solve the following problem using this method: A school plans to build a cylindrical water tower as shown in Figure 2. $$AB$$ is one of the generatrices, and $$C$$ is the midpoint of $$AB$$. The radius $$R$$ of the base circle and the height $$h$$ of the cylinder satisfy $$R=\dfrac{\sqrt{h}}{ \pi }$$. Now, it is planned to build an iron ladder from point $$A$$ to point $$C$$ around the surface of the cylinder to observe the operation of the water tower, and it takes 1 full turn to reach from $$A$$ to $$C$$. If $$h=16$$ meters, then the length of the iron ladder is ___ meters.
|
$$8\sqrt{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/98ff0ce1-417e-4208-80b0-46951446b3ec.png"
}
] |
8056636c-fa77-4297-a053-0fdb81a0be02
|
<image>As shown in the figure, the base edge length of a regular triangular prism is $$4$$. A plane passing through $$BC$$ forms a $$30^{\circ}$$ dihedral angle with the base. This plane intersects the lateral edge $$AA_{1}$$ at point $$D$$. Then the length of $$AD$$ is ___.
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8056636c-fa77-4297-a053-0fdb81a0be02.png"
}
] |
f1022e47-5b22-4d5b-a118-f0d34f3e7daf
|
<image>There is a number array as follows: Let the $$j$$th number in the $$i$$th row be $$a_{ij}$$ (for example, $$a_{43}=19$$), then $$a_{47}-a_{65}=$$ ___.
|
$$-2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f1022e47-5b22-4d5b-a118-f0d34f3e7daf.png"
}
] |
bf910a52-bb77-4869-a0c2-d518d652a9b1
|
<image>If the shape and related data seen from the left side and top of a rectangular prism are as shown in the figure, then the area of the shape seen from the front is ______.
|
50
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bf910a52-bb77-4869-a0c2-d518d652a9b1.png"
}
] |
bcf384d9-4acd-4455-af15-907cc81e07c1
|
<image>Execute the program as shown in the figure. If the input is $$m=98$$, $$n=63$$, then the output $$m=$$______．
|
$$7$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bcf384d9-4acd-4455-af15-907cc81e07c1.png"
}
] |
243c98b2-86aa-4892-baeb-82e62d005dd0
|
<image>Execute the program flowchart as shown in the figure, the sum of all output values is ______.
|
$$48$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/243c98b2-86aa-4892-baeb-82e62d005dd0.png"
}
] |
5ace095a-6f8f-4228-b262-5d71757311cb
|
<image>There is a numerical converter, the process is shown in the figure. If the initial input value of $$x$$ is $$10$$, it can be found that the result of the first output is $$5$$. The result of the second output is $$-2$$, and so on. What is the result of the $$2013$$th output?
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5ace095a-6f8f-4228-b262-5d71757311cb.png"
}
] |
ddd0143a-0bce-4b59-b401-403b079a5e08
|
<image>The system of linear inequalities that can represent the shaded area in the figure is ___.
|
$$\begin{cases} x \leqslant 0, 0 \leqslant y \leqslant 1, 2x-y+2 \geqslant 0 \end{cases}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ddd0143a-0bce-4b59-b401-403b079a5e08.png"
}
] |
69a5c8c2-16c8-47ca-9b0f-fb2cd5ad7a0c
|
<image>When the pattern shown in the figure is folded to form a cube, the number ______ will be on the face opposite the face with the number 2.
|
5
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/69a5c8c2-16c8-47ca-9b0f-fb2cd5ad7a0c.png"
}
] |
bb4b6d2e-9a12-4147-a437-5ce26cd07372
|
<image>The cross-section of a corner slot is shown in the figure. Quadrilateral $$ADEB$$ is a rectangle. If $$α=50^{\circ}$$, $$β=70^{\circ}$$, $$AC=90mm$$, and $$BC=150mm$$, then the length of $$DE$$ is ___ $$mm$$.
|
$$210$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bb4b6d2e-9a12-4147-a437-5ce26cd07372.png"
}
] |
68830015-c06b-4266-937a-c5a0e2fcc58c
|
<image>As shown in the figure, in $$\triangle ABC$$, $$DE \parallel FG \parallel BC$$, and $$AD:DF:FB=1:2:3$$. If $$EG=3$$, then $$AC=$$ ___.
|
$$9$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/68830015-c06b-4266-937a-c5a0e2fcc58c.png"
}
] |
a2ac0f66-0404-4e49-a2f6-8d8e21752004
|
<image>As shown in the figure, lines $$l_{1}$$ and $$l_{2}$$ are cut by line $$l_{3}$$. If $$l_{1} \parallel l_{2}$$ and $$\angle 1 = 48^{\circ}$$, then $$\angle 2 =$$ ___ $$\unit{^{\circ}}$$.
|
$$132$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a2ac0f66-0404-4e49-a2f6-8d8e21752004.png"
}
] |
75583e95-f4de-4aba-94f3-0f8408decedd
|
<image>As shown in the figure, the line l1: y = x + 1 intersects the line 32 at point P(a, 2). The solution to the system of equations {y = x + 1, y = mx + n} is ______.
|
{x=1, y=2}
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/75583e95-f4de-4aba-94f3-0f8408decedd.png"
}
] |
0a332c0e-150c-4cd0-9bca-1b44e7e5f340
|
<image>In the production process of a certain product at a factory, the corresponding data for production volume $$x$$ (tons) and energy consumption $$y$$ (tons) are shown in the table below. According to the method of least squares, the regression line equation is $$\hat{y}=0.7x+a$$. When the production volume is $$80$$ tons, the estimated energy consumption is ___ tons.
|
$$59.5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0a332c0e-150c-4cd0-9bca-1b44e7e5f340.png"
}
] |
dd8dd64b-883c-480b-99d1-1fdf96c7f829
|
<image>Execute the program flowchart shown in the figure, the output result is $$S=$$______．
|
$$1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/dd8dd64b-883c-480b-99d1-1fdf96c7f829.png"
}
] |
395beba7-2375-4239-9e27-3221751d71dd
|
<image>Arrange consecutive positive integers according to the following pattern, then the number $$x$$ located at the 7th row and 7th column is ______.
|
$$85$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/395beba7-2375-4239-9e27-3221751d71dd.png"
}
] |
1b02c737-5a0b-444d-aa83-f1e28a132e95
|
<image>As shown in the figure, there is some water in a cubic container with an edge length of 10 cm, and the water height is 7 cm. A rectangular iron block, 8 cm in height, is placed vertically into the water (the base of the iron block is parallel to the bottom of the container). Before the iron block is completely submerged, the water is already full, and the submerged part of the iron block is 6 cm high. The volume of this iron block is ______ cubic centimeters.
|
400
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1b02c737-5a0b-444d-aa83-f1e28a132e95.png"
}
] |
acf1e8cb-5531-4a45-9046-f15d271a8606
|
<image>There is a clip as shown in the figure, $$AB=2BC$$, $$BD=2BE$$, in front of the clip there is a rectangular solid object, with thickness $$PQ$$ being $$6cm$$. If you want to use the tips of the clip $$A$$ and $$D$$ to hold points $$P$$ and $$Q$$, then the part where you hold the clip $$EC$$ must open at least ______ $$cm$$.
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/acf1e8cb-5531-4a45-9046-f15d271a8606.png"
}
] |
ca227f36-1b51-40f2-aa73-a47c58f82c23
|
<image>In an isosceles right triangle $$\text{Rt}\triangle ABC$$, with the right angle at vertex $$A$$, a circle is drawn with $$A$$ as the center and the altitude $$AD$$ as the radius, intersecting $$AB$$ and $$AC$$ at points $$E$$ and $$F$$ respectively. Given that $$AD = 2$$ cm, the area of the shaded region in the figure is ___ square centimeters.
|
$$0.86$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ca227f36-1b51-40f2-aa73-a47c58f82c23.png"
}
] |
7b6d334e-9645-4681-be0b-6a5a06b66689
|
<image>As shown in the figure, line segments $$AD$$ and $$BC$$ intersect at point $$O$$, and $$AB \parallel CD$$. If $$AB:CD=2:3$$ and the area of $$\triangle ABO$$ is $$2$$, then the area of $$\triangle CDO$$ is ___.
|
$$\dfrac{9}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7b6d334e-9645-4681-be0b-6a5a06b66689.png"
}
] |
9e35b0f1-bd61-4aab-8d15-5b14497d4c61
|
<image>As shown in the figure, the diagonals AC and BD of rhombus ABCD intersect at point O, and E is the midpoint of AD. If OE = 5 and BD = 12, then the area of rhombus ABCD is.
|
96
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9e35b0f1-bd61-4aab-8d15-5b14497d4c61.png"
}
] |
cfb05725-d9e4-49fc-8582-41534c841fd6
|
<image>As shown in the figure, to measure the height of the flagpole $AB$ on the opposite bank of the river, the angle of elevation to the top of the flagpole $A$ from point $C$ is $30^\circ$. Moving $5$ meters in the direction of $CB$ to point $D$, the angle of elevation to the top of the flagpole $A$ from point $D$ is $45^\circ$. The height of the flagpole $AB$ is meters.
|
$\frac{5\left( 1+\sqrt{3} \right)}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cfb05725-d9e4-49fc-8582-41534c841fd6.png"
}
] |
47c58cda-1097-4254-84f1-4cfdede2b870
|
<image>In the right triangle ABC, ∠BAC = 90°, AD is the median of side BC, ED is perpendicular to BC at D, and intersects the extension of BA at E. If ∠E = 40°, then the degree measure of ∠BDA is
|
80°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/47c58cda-1097-4254-84f1-4cfdede2b870.png"
}
] |
0bf06a66-b608-4398-94dd-6dce5bb190cd
|
<image>As shown in the figure, ∠AOC = 40°, OD bisects ∠AOB, OE bisects ∠BOC, then the measure of ∠DOE is.
|
20°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0bf06a66-b608-4398-94dd-6dce5bb190cd.png"
}
] |
5d34d4b8-2585-41ad-9ee8-871b8ba7c879
|
<image>As shown in the figure, from point B at the top of building AB, the angle of depression to the bottom C of another building CD is 60 degrees. It is known that the distance between points A and C is 15 meters. What is the height of building AB? (Express the result with a square root)
|
$15\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5d34d4b8-2585-41ad-9ee8-871b8ba7c879.png"
}
] |
e131a51a-4fcd-44c8-b8ce-bfad304ce8c8
|
<image>Carefully observe the shapes represented by the following 4 numbers: Question: What is the number of small squares in the shape represented by the number 100?
|
20201
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e131a51a-4fcd-44c8-b8ce-bfad304ce8c8.png"
}
] |
a050f36a-42ee-430f-ba64-1e534ac4b207
|
<image>A light bulb (considered as a point) is located directly above a circular tabletop (with a circular hole of 0.4 m in diameter at the center). The light from the bulb casts a shadow of the tabletop on the ground, forming a ring-shaped shadow as shown in the figure. Given that the diameter of the tabletop is 1.2 m, the tabletop is 1 m above the ground, and the light bulb is 3 m above the ground, the area of the ring-shaped shadow on the ground is m$^{2}$.
|
0.72π
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a050f36a-42ee-430f-ba64-1e534ac4b207.png"
}
] |
49de088b-62d6-44ba-b860-807f434facb9
|
<image>In the figure, in $\Delta ABC$, point $D$ is the intersection of the angle bisectors of $\angle ABC$ and $\angle ACB$, $\angle ABC=60{}^\circ $, $\angle ACB=40{}^\circ $. Then $\angle BDC$ is.
|
130°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/49de088b-62d6-44ba-b860-807f434facb9.png"
}
] |
5b71847f-bb31-47fc-b80d-1a1746954e04
|
<image>As shown in the figure, in the square grid, △ABC, if the side length of the small square is 1, then the area of △ABC is.
|
5
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5b71847f-bb31-47fc-b80d-1a1746954e04.png"
}
] |
6b246cf0-d4ab-4650-86fd-9d3f99467ec2
|
<image>In the figure, $AB$ is a chord of circle $O$, and $OC \bot AB$ at point $C$. If $AB=8$ and $OC=4$, then the radius of circle $O$ is.
|
$4\sqrt{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6b246cf0-d4ab-4650-86fd-9d3f99467ec2.png"
}
] |
002a616e-8782-4425-902a-0c3e01ae0c62
|
<image>In $\Delta ABC$, $\angle BAC=60{}^\circ $, $AB=4$, $AC=6$, $\overrightarrow{AB}=2\overrightarrow{AD}$, $\overrightarrow{AE}=2\overrightarrow{EC}$, $\overrightarrow{EF}=2\overrightarrow{FD}$. Find the value of $\overrightarrow{BF}\cdot \overrightarrow{DE}$.
|
4;
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/002a616e-8782-4425-902a-0c3e01ae0c62.png"
}
] |
d39230da-766b-4647-a115-ddb2be08ef77
|
<image>Given: As shown in the figure, points A and B are on the edge of a dihedral angle of $60{}^\circ$. Lines AC and BD are in the two half-planes of the dihedral angle and are perpendicular to AB. It is known that $AB=4$, $AC=6$, and $BD=8$. Find $CD$.
|
$2\sqrt{17}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d39230da-766b-4647-a115-ddb2be08ef77.png"
}
] |
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