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c880c117-9e9c-4ac2-8a9d-63b516ab4c95
|
<image>As shown in the figure, AC = BC, DC = EC, ∠ACB = ∠ECD = 90°, and ∠EBD = 62°. Find ∠AEB.
|
152°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c880c117-9e9c-4ac2-8a9d-63b516ab4c95.png"
}
] |
9897501e-6b85-4ff3-a85a-ff78919dcd07
|
<image>As shown in the figure, in parallelogram ABCD, AB=1, ∠BAD=120°. Connect BD, and draw AE//BD to intersect the extension of CD at point E. Draw EF⊥BC to intersect the extension of BC at point F. What is the length of EF?
|
$\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9897501e-6b85-4ff3-a85a-ff78919dcd07.png"
}
] |
0b0dc1c0-8c8d-400f-82f4-0d19d4d0b639
|
<image>As shown in the figure, in the right rectangular prism $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$, the base $ABCD$ is a square, and $A{{A}_{1}}=\sqrt{3}AB$. Let the angle formed by the skew lines $A{{B}_{1}}$ and $BD$ be $\theta$, then $\cos \theta =$.
|
$\frac{\sqrt{2}}{4}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0b0dc1c0-8c8d-400f-82f4-0d19d4d0b639.png"
}
] |
5c341bee-3a2a-472f-a253-b7ed703c8e37
|
<image>Xiaogang rides his bike to school from home, first going uphill to location A and then downhill to the school, as shown in the graph. If the speeds going uphill and downhill remain the same on the return trip, how many minutes will it take Xiaogang to get back home from school?
|
12
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5c341bee-3a2a-472f-a253-b7ed703c8e37.png"
}
] |
ffb19cd2-08a9-44f9-bd55-cc59e1dc3a9d
|
<image>As shown in the figure, quadrilateral EFGH is an inscribed square in △ABC, with BC = 21 cm and height AD = 15 cm. Then the side length of the inscribed square EF =.
|
8.75 cm
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ffb19cd2-08a9-44f9-bd55-cc59e1dc3a9d.png"
}
] |
655cefcc-904d-484f-b6a0-95593e1b1147
|
<image>The diagram below shows a 'numerical transformation machine'. If the input values are $x=-\frac{3}{2}$ and $y=2$, then the output value is.
|
$-\frac{5}{4}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/655cefcc-904d-484f-b6a0-95593e1b1147.png"
}
] |
22101f9b-8a6d-4f87-9368-c762000cf513
|
<image>As shown in the figure, ⊙O is the circumcircle of △ABC, ∠BAC = 60°. If the radius OC of ⊙O is 2, then the length of chord BC is.
|
$2\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/22101f9b-8a6d-4f87-9368-c762000cf513.png"
}
] |
c1572ee4-b08e-4b07-8783-eb1fde2e64c2
|
<image>As shown in the figure, M is the midpoint of AB in parallelogram ABCD, and CM intersects BD at E. The ratio of the area of the shaded part to the area of parallelogram ABCD is:
|
1:3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c1572ee4-b08e-4b07-8783-eb1fde2e64c2.png"
}
] |
18592ab9-6267-4d53-a0cd-de9ecb3568e1
|
<image>In the figure, in $\Delta ABC$, it is known that points $D$, $E$, and $F$ are the midpoints of $BC$, $AD$, and $CE$ respectively, and ${{S}_{\Delta ABC}}=16\text{cm}^{\text{2}}$. Find ${{S}_{\Delta BEF}}=$$\text{cm}^{\text{2}}$.
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/18592ab9-6267-4d53-a0cd-de9ecb3568e1.png"
}
] |
42fb2c66-0d35-4374-ad20-b5d585a22bdf
|
<image>As shown in the figure, the perimeter of the triangle is 24 cm, and the area is 24 square cm. What is the area of the circle in square cm?
|
4π
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/42fb2c66-0d35-4374-ad20-b5d585a22bdf.png"
}
] |
245471b8-638a-4771-9998-1a28516c1226
|
<image>In △ABC, CD is perpendicular to AB at point D, BE is perpendicular to AC at point E, F is the midpoint of BC, DE = 5, BC = 8, then the perimeter of △DEF is.
|
13
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/245471b8-638a-4771-9998-1a28516c1226.png"
}
] |
02a14e0c-e8b6-43fb-a5b8-847312e9978c
|
<image>Given that the points corresponding to the numbers $a$ and $b$ on the number line are as shown in the figure, simplify the expression $\left| {{a}^{2}}+b \right|+\left| {{a}^{2}}-b \right|$.
|
2b
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/02a14e0c-e8b6-43fb-a5b8-847312e9978c.png"
}
] |
45c85311-595b-4992-b9a6-093ae3b5e230
|
<image>A butterfly randomly lands on one of the small triangles within the rectangular window glass as shown in the figure, then the probability that the butterfly lands exactly on the white area is.
|
$\frac{1}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/45c85311-595b-4992-b9a6-093ae3b5e230.png"
}
] |
e39e03e6-a814-4104-8a63-1a1b321515fc
|
<image>In the figure, AB=AC, OB=OC, ∠BAO=25º, then ∠CAO=
|
25°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e39e03e6-a814-4104-8a63-1a1b321515fc.png"
}
] |
1d1c2e1b-fb71-4e90-b33d-fa0f1bb1732b
|
<image>Given that the length of line segment AB is 12, M is the midpoint of line segment AB. If point C divides line segment MB into MC:CB = 1:2, then the length of line segment AC is
|
8
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1d1c2e1b-fb71-4e90-b33d-fa0f1bb1732b.png"
}
] |
043add0c-eeac-4deb-b8f3-41105527f562
|
<image>As shown in the figure, in the Cartesian coordinate system, the square ABCD with an area of 100 has two vertices A and B sliding on the coordinate axes. Point B starts from the origin O and moves along the positive x-axis, while point A moves along the positive y-axis towards the origin O. When ∠ABO = 40°, the length of the path traced by the midpoint E of side AB is.
|
$\frac{25\pi }{18}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/043add0c-eeac-4deb-b8f3-41105527f562.png"
}
] |
a1dccc3f-c97a-4bb0-b49f-79973dfc6247
|
<image>As shown in the figure, it is known that $\angle 1={{\left( 2x+10 \right)}^{\text{o}}}$, $\angle 2\text{=6}{{\text{0}}^{\text{o}}}$, $\angle 3\text{=}{{\left( 2x-10 \right)}^{\text{o}}}$, then $\angle \text{1}=\text{ }$°.
|
70
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a1dccc3f-c97a-4bb0-b49f-79973dfc6247.png"
}
] |
2927b88f-51e9-45da-bfe4-af8818edda74
|
<image>As shown in Figure 1, in a square ABCD with side length 4 cm, point P moves from point A at a speed of 2 cm per second along the path AB→BC, stopping at point C. A line PQ is drawn parallel to BD, intersecting side AD (or side CD) at point Q. The graph in Figure 2 shows the relationship between the length y (cm) of PQ and the time x (seconds) of point P's movement. When point P has been moving for 2.5 seconds, the length of PQ is cm.
|
$3\sqrt{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2927b88f-51e9-45da-bfe4-af8818edda74.png"
}
] |
9e51f6f1-21ef-4c98-9c30-6b2327f89f5c
|
<image>A school plans to build a rectangular swimming pool with an area of 392m², and construct a path 2m and 4m wide around it (as shown in the figure). What is the minimum area in m² that the construction will occupy?
|
648
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9e51f6f1-21ef-4c98-9c30-6b2327f89f5c.png"
}
] |
e524b39b-0acb-42f8-96be-eb94f99f2e18
|
<image>In a regular tetrahedron $P-ABC$, D and E are the midpoints of AB and BC, respectively. There are the following three statements: $1. AC \bot PB$; $2. AC \parallel$ plane PDE; $3. AB \bot$ plane PDE. The number of correct statements is.
|
2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e524b39b-0acb-42f8-96be-eb94f99f2e18.png"
}
] |
ae85450b-6f94-46e6-966a-d4e4b0e00a64
|
<image>As shown in the figure, in $$\triangle ABC$$, $$AB=6$$, $$AC=10$$, points $$D$$, $$E$$, and $$F$$ are the midpoints of $$AB$$, $$BC$$, and $$AC$$, respectively. What is the perimeter of quadrilateral $$ADEF$$?
|
$$16$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ae85450b-6f94-46e6-966a-d4e4b0e00a64.png"
}
] |
fe2d4abb-cd96-44d3-a814-a5a497f99baa
|
<image>As shown in the figure, in $$\triangle ABC$$, $$CD$$ bisects $$\angle ACB$$ and intersects $$AB$$ at point $$D$$, $$DE \bot AC$$ intersects at point $$E$$, $$DF \bot BC$$ intersects at point $$F$$, and $$BC = 4$$, $$DE = 2$$. What is the area of $$\triangle BCD$$?
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/fe2d4abb-cd96-44d3-a814-a5a497f99baa.png"
}
] |
b9d0e528-7b47-4901-a845-d50adf4f630d
|
<image>Given three points $$A$$, $$B$$, and $$C$$ on a number line representing the numbers $$-2$$, $$-0.5$$, and $$3$$ respectively. If a point is chosen at random on the line segment $$AC$$, what is the probability that the distance from this point to point $$B$$ is no more than $$1$$?
|
$$\dfrac{2}{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b9d0e528-7b47-4901-a845-d50adf4f630d.png"
}
] |
88b97f26-89c4-4d6e-9237-fb48a7a386a3
|
<image>In the figure, in $$\triangle ABC$$, $$\angle A=30^{\circ}$$, $$\angle B=45^{\circ}$$, $$AC=2\sqrt{3}$$, then the length of $$AB$$ is ___.
|
$$3+\sqrt{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/88b97f26-89c4-4d6e-9237-fb48a7a386a3.png"
}
] |
330a20dc-c92a-4098-bddb-3d137c58a3bc
|
<image>As shown in the figure, quadrilateral $$ABCD$$ is an inscribed quadrilateral in circle $$O$$. Extend $$AB$$ and $$DC$$ to intersect at point $$P$$. If $$\dfrac{PB}{PD}=\dfrac{1}{3}$$, then the value of $$\dfrac{BC}{AD}$$ is ___.
|
$$\dfrac{1}{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/330a20dc-c92a-4098-bddb-3d137c58a3bc.png"
}
] |
f5d9664e-1894-45d3-9a5c-740a9e494339
|
<image>In a rectangle, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. If $$AO=1$$, then $$BD=$$___.
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f5d9664e-1894-45d3-9a5c-740a9e494339.png"
}
] |
07414a7d-8bf7-4fa2-95a1-c9da48293641
|
<image>As shown in the figure, plane $$\alpha \parallel$$ plane $$\beta$$, point $$A$$, $$C \in \alpha$$, point $$B$$, $$D \in \beta$$, $$AB=a$$ is the common perpendicular of $$\alpha$$ and $$\beta$$, and $$CD$$ is a slant line. If $$AC=BD=b$$, $$CD=c$$, and $$M$$, $$N$$ are the midpoints of $$AB$$ and $$CD$$ respectively, then the length of $$MN$$ is ___.
|
$$\dfrac{1}{2}\sqrt{4b^{2}+a^{2}-c^{2}}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/07414a7d-8bf7-4fa2-95a1-c9da48293641.png"
}
] |
0c186609-6f61-45bf-b246-370741b80bfa
|
<image>As shown in the figure, in $$\triangle ABC$$, points $$D$$ and $$E$$ are on sides $$AB$$ and $$AC$$ respectively, with $$DE \parallel BC$$, $$AD = 10$$, $$BD = 5$$, and $$AE = 6$$. The length of $$CE$$ is ___.
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0c186609-6f61-45bf-b246-370741b80bfa.png"
}
] |
2b89ff4c-a422-4c9a-86e9-55f39f32760c
|
<image>Execute the program shown in the figure, after inputting a=3, b=-1, n=4, the output result is _____.
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2b89ff4c-a422-4c9a-86e9-55f39f32760c.png"
}
] |
f935636d-d79b-4abf-9254-842e6fc3fede
|
<image>As shown in the figure, in circle $$ \odot O$$, $$AB$$ is a chord of $$ \odot O$$, and radius $$OC \perp AB$$ at point $$D$$. If the length of $$OB$$ is $$10$$, and $$ \sin \angle BOD=\dfrac{4}{5}$$, then the length of $$AB$$ is ___.
|
$$16$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f935636d-d79b-4abf-9254-842e6fc3fede.png"
}
] |
be9adad4-5d43-4ae2-9759-47a92a81800e
|
<image>The figure shows a plane net of a cube. The minimum sum of the numbers on the opposite faces of the original cube is ___.
|
$$6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/be9adad4-5d43-4ae2-9759-47a92a81800e.png"
}
] |
76301ab1-ee35-4179-9d8b-6800d3b26d71
|
<image>As shown in the figure, in $$\triangle ABC$$, $$D$$ is the midpoint of $$BC$$. Given that $$\overrightarrow{AB}=\overrightarrow{a}$$ and $$\overrightarrow{AC}=\overrightarrow{b}$$, then $$\overrightarrow{BD}=$$ ___.
|
$$\dfrac{1}{2}\left(\overrightarrow{b}-\overrightarrow{a}\right)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/76301ab1-ee35-4179-9d8b-6800d3b26d71.png"
}
] |
61b646f3-0980-43f3-8d14-15a531f2913a
|
<image>Given that the quadrilateral is a square, the area of the blank triangle is $$56$$ square centimeters, and the length of $$ED$$ is $$7$$ centimeters, find the area of the shaded part ______ square centimeters.
|
200
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/61b646f3-0980-43f3-8d14-15a531f2913a.png"
}
] |
86f61abd-1888-4096-bb11-d9b16a952a5b
|
<image>Use black and white regular hexagonal floor tiles to form several patterns according to the rule shown in the figure. Then, the number of white floor tiles in the $$n$$th pattern is ______.
|
$$(4n+2)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/86f61abd-1888-4096-bb11-d9b16a952a5b.png"
}
] |
c0710266-e3d2-4939-a582-80550e79f9e2
|
<image>As shown in the figure, $$PO\bot $$ plane $$ABC$$, $$BO\bot AC$$. In the figure, the number of lines perpendicular to $$AC$$ is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c0710266-e3d2-4939-a582-80550e79f9e2.png"
}
] |
f4d8b618-aa32-4ffc-b506-d05e7239e870
|
<image>As shown in the figure, in the triangular prism $$ABC-A_{1}B_{1}C_{1}$$, $$AB:A_{1}B_{1}=1:2$$. The volume ratio of the tetrahedrons $$A_{1}-ABC$$, $$B-A_{1}B_{1}C$$, and $$C-A_{1}B_{1}C_{1}$$ is ___.
|
$$1:2:4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f4d8b618-aa32-4ffc-b506-d05e7239e870.png"
}
] |
83e22fc5-15a5-482f-b018-37e705c571e9
|
<image>As shown in the figure, PAB and PCD are two secant lines of circle O. If PA=5, AB=7, and CD=11, then AC:BD=．
|
$\frac{1}{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/83e22fc5-15a5-482f-b018-37e705c571e9.png"
}
] |
f9d4854f-e753-44c4-bcab-330dc538b245
|
<image>As shown in the figure, three identical small rectangles are arranged in a 'vertical-horizontal-vertical' pattern within a larger rectangle that is 10 units long and 8 units wide. What is the width of one of the small rectangles?
|
2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f9d4854f-e753-44c4-bcab-330dc538b245.png"
}
] |
86fda4a9-6d65-404f-be57-60935463b5a2
|
<image>As shown in the figure, the height of the reading pavilion $MN$ at a certain corner of a school is $\left( 30-10\sqrt{3} \right)m$. There is a decorative light tower $PQ$ in the exact east direction of the reading pavilion. At point $A$ on the ground between them (points $M$, $A$, and $P$ are collinear), the angles of elevation to the top of the reading pavilion $N$ and the top of the light tower $Q$ are $15^\circ$ and $60^\circ$ respectively. From the top of the reading pavilion $N$, the angle of elevation to the top of the light tower $Q$ is $30^\circ$. What is the height of the light tower $PQ$ in meters?
|
60
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/86fda4a9-6d65-404f-be57-60935463b5a2.png"
}
] |
58bfcaa2-7eee-4967-911d-d0d34ee7e7f5
|
<image>As shown in the figure, 1000 beans are randomly scattered in a square with a side length of 1. 380 beans fall on the shaded red heart area. Estimate the area of the shaded red heart area.
|
0.38
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/58bfcaa2-7eee-4967-911d-d0d34ee7e7f5.png"
}
] |
e8610d83-8da6-4277-a294-fb7f0a2c93d2
|
<image>As shown in the figure, 9 identical smaller rectangles are placed inside rectangle ABCD, with relevant data indicated in the diagram, then the area of the shaded part in the figure is (square units).
|
18
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e8610d83-8da6-4277-a294-fb7f0a2c93d2.png"
}
] |
b6873114-f115-4b44-a740-6d869e9b5cc1
|
<image>Given the function $f\left( x \right)=A\sin \left( \omega x+\varphi \right)$ (where $A, \omega, \varphi$ are parameters, and $A > 0, \omega > 0, 0 < \varphi < \pi$), part of its graph is shown in the figure. What is the value of $\varphi$?
|
$\frac{\pi }{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b6873114-f115-4b44-a740-6d869e9b5cc1.png"
}
] |
d4d25cc5-fb63-4572-84eb-0f8ef02b1a5f
|
<image>As shown in the figure, △ABD and △BCE are both isosceles right triangles. If CD=8 and BE=3, then AC equals
|
$\sqrt{34}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d4d25cc5-fb63-4572-84eb-0f8ef02b1a5f.png"
}
] |
bea8052f-fdb9-46c3-83f4-66aaff7623b7
|
<image>In the operation program shown in the figure, if the initial input value of $x$ is 12, we find that the result of the 1st output is 6, the result of the 2nd output is 3, ..., the result of the 2020th output is.
|
12;
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bea8052f-fdb9-46c3-83f4-66aaff7623b7.png"
}
] |
ccbaf894-299d-4a08-9a72-b6cd27521977
|
<image>The speed of light varies in different media, so when light travels from water to air, it refracts. Because the refractive index is the same, parallel light rays in water remain parallel in air. Given EF∥AB∥CD, ∠2=3∠3, ∠8=2∠5+10°, what is the result of ∠7-∠4 in degrees?
|
28
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ccbaf894-299d-4a08-9a72-b6cd27521977.png"
}
] |
92a2370e-f3d8-4744-a5c1-b34bdcaf5ca7
|
<image>Execute the program flowchart shown in the figure, then the output value of $x$ is.
|
14
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/92a2370e-f3d8-4744-a5c1-b34bdcaf5ca7.png"
}
] |
e392636f-a33c-4354-8ce2-c7eaa91f7e58
|
<image>As shown in the figure: A right-angled isosceles triangle is connected to a square, and another square is connected to the leg of the right-angled isosceles triangle, and so on, forming a tree-like shape known as a 'Pythagorean Tree'. If a certain Pythagorean Tree contains 1023 squares, and the side length of the largest square is $\frac{\sqrt{2}}{2}$, then the side length of the smallest square is.
|
$\frac{1}{32}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e392636f-a33c-4354-8ce2-c7eaa91f7e58.png"
}
] |
3dd03b54-3999-4939-b421-20490bbd1ba9
|
<image>As shown in the figure, in $\vartriangle ABC$, $\angle B=90{}^\circ $, $\angle A=60{}^\circ $, $BC=5$. If $\vartriangle ABC$ is translated 2 units to the right along the line containing the right side $BC$ to reach $\vartriangle DEF$, and $AC$ intersects $DE$ at point $G$, then the length of $EG$ is.
|
$\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3dd03b54-3999-4939-b421-20490bbd1ba9.png"
}
] |
83593dcc-aef0-43f9-9f89-b382eda96fd7
|
<image>In the figure, in △ABC, ∠A = 20°, the angle bisectors of ∠ABC and ∠ACB intersect at point D₁, the angle bisectors of ∠ABD₁ and ∠ACD₁ intersect at point D₂, and so on, the angle bisectors of ∠ABD₂ and ∠ACD₂ intersect at point D. What is the measure of ∠BDC?
|
40°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/83593dcc-aef0-43f9-9f89-b382eda96fd7.png"
}
] |
fee0454e-6173-415c-9d4b-69c679a7ed9b
|
<image>As shown in the figure, arcs are drawn within a square using the square's four vertices and its center as the centers, and with a radius equal to half the side length of the square. If a point is chosen at random within the square, the probability that the point falls within the shaded area is.
|
$\frac{\pi }{2}-1$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/fee0454e-6173-415c-9d4b-69c679a7ed9b.png"
}
] |
f6f6d9a0-52a0-49d9-9568-22994b6235d1
|
<image>As shown in the figure, $\vartriangle FDE$ is obtained from $\vartriangle ABC$ through a certain transformation. Observe the relationship between the coordinates of point $A$ and point $E$, and point $B$ and point $D$. If a point $M$ on $\vartriangle ABC$ has coordinates $(x,\,y)$, then the coordinates of its corresponding point on $\vartriangle FDE$ are.
|
$(-x,\,-y)$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f6f6d9a0-52a0-49d9-9568-22994b6235d1.png"
}
] |
a1b6b5de-33b8-4975-ba6d-5aea90cb3d7e
|
<image>As shown in the figure, in square ABCD, M and N are the midpoints of BC and CD, respectively. If $\overrightarrow{AC}=\lambda \overrightarrow{AM}+\mu \overrightarrow{BN}$, then $\lambda +\mu =$.
|
$\frac{8}{5}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a1b6b5de-33b8-4975-ba6d-5aea90cb3d7e.png"
}
] |
55017d51-cd04-4505-9341-87241b7671de
|
<image>The following shapes are all composed of circles and equilateral triangles of the same size according to a certain pattern. The 1st shape is composed of 8 circles and 1 equilateral triangle, the 2nd shape is composed of 16 circles and 4 equilateral triangles, the 3rd shape is composed of 24 circles and 9 equilateral triangles, … then in which shape do the number of circles and equilateral triangles become equal?
|
8
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/55017d51-cd04-4505-9341-87241b7671de.png"
}
] |
340385bf-e6fa-4ccb-9095-17a70a7a1879
|
<image>Given the function $f(x) = \sin(\omega x + \varphi) (\omega > 0, |\varphi| < \frac{\pi}{2})$, part of its graph is shown in the figure below, then $\varphi =$.
|
$-\frac{\pi}{6}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/340385bf-e6fa-4ccb-9095-17a70a7a1879.png"
}
] |
e5f464e3-4b61-4aff-a8d1-ac255e290400
|
<image>The figure below is a net of a cube. If the net is folded into a cube such that the sum of the numbers on opposite faces is 5, then what is the value of $x+y+z$?
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e5f464e3-4b61-4aff-a8d1-ac255e290400.png"
}
] |
7d5ba252-0272-4101-99ee-ee82ae692d9e
|
<image>As shown in the figure, line $$EO \bot BC$$ at point $$O$$, $$\angle BOC = 3\angle 1$$, $$OD$$ bisects $$\angle AOC$$, then the measure of $$\angle 2$$ is ___.
|
$$30^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7d5ba252-0272-4101-99ee-ee82ae692d9e.png"
}
] |
333f4e7a-492c-4193-8d93-c6ad4550825b
|
<image>A school in this city has launched an educational campaign themed 'Promote Green Travel, Care for the Health of Teachers and Students.' To understand the travel methods of teachers and students in the school, a random sample of some teachers and students was conducted, and the collected data was plotted into the following two incomplete statistical charts. It is known that the number of teachers randomly surveyed is half the number of students. According to the information in the charts, the number of teachers who travel by private car is ___.
|
$$15$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/333f4e7a-492c-4193-8d93-c6ad4550825b.png"
}
] |
d1988e02-5037-4f63-a8f0-a615d2d2bb3f
|
<image>For any non-zero real numbers a, b, if the operation principle of a\bigotimes b is as shown in the figure, then log_{2}8\bigotimes(\frac{1}{2})^{-2} =_____.
|
1
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d1988e02-5037-4f63-a8f0-a615d2d2bb3f.png"
}
] |
74a1aa14-dc11-4369-acb0-15976ab43bf2
|
<image>Given the function $$f(x)=\sin ( \omega x+ \varphi )( \omega > 0,- \pi < \varphi < 0)$$, part of its graph is shown in the figure. Then $$f\left(\dfrac{\varphi }{\omega } \right)=$$ ___.
|
$$-1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/74a1aa14-dc11-4369-acb0-15976ab43bf2.png"
}
] |
6cd48212-d0fc-4cad-bc18-7d4335b6d680
|
<image>The diagram below shows Xiao Ming using a laser instrument on building $$AB$$ to measure the height of another building $$CD$$. On the ground at point $$P$$, a plane mirror is placed horizontally. A laser beam is emitted from point $$A$$, reflects off point $$P$$ on the mirror, and precisely hits the top point $$C$$ of building $$CD$$. It is known that $$AB \perp BD$$, $$CD \perp BD$$, and the measurements are $$AB=15\ \unit{m}$$, $$BP=20\ \unit{m}$$, $$PD=32\ \unit{m}$$, with points $$B$$, $$P$$, and $$D$$ lying on a straight line. The height of building $$CD$$ is ___$$\unit{m}$$.
|
$$24$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6cd48212-d0fc-4cad-bc18-7d4335b6d680.png"
}
] |
b27339c8-7ee8-4e31-9ab3-8554189fc867
|
<image>As shown in the figure, the side length of the regular hexagon $$A_{1}B_{1}C_{1}D_{1}E_{1}F_{1}$$ is $$1$$. The six diagonals form another regular hexagon $$A_{2}B_{2}C_{2}D_{2}E_{2}F_{2}$$, and this process continues. What is the area of the regular hexagon $$A_{4}B_{4}C_{4}D_{4}E_{4}F_{4}$$?
|
$$\dfrac{\sqrt{3}}{18}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b27339c8-7ee8-4e31-9ab3-8554189fc867.png"
}
] |
722c72b0-d3bf-4a63-85a9-5f1baa6a76be
|
<image>A square paper strip is folded as shown in the figure. If the resulting $$\angle AOB' = 40^{\circ}$$, then $$\angle B'OG =$$ ___.
|
$$70^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/722c72b0-d3bf-4a63-85a9-5f1baa6a76be.png"
}
] |
94305af6-d0a6-40e8-b750-3188c5b34910
|
<image>As shown in the figure, given AB⊥BC, CD=33, ∠ACB=30°, ∠BCD=75°, ∠BDC=45°, then AB=_____.
|
11\sqrt{2}
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/94305af6-d0a6-40e8-b750-3188c5b34910.png"
}
] |
f65b400a-0cc1-42c5-b127-9a8dc8f0f5bd
|
<image>Read the flowchart shown below and write the function it represents.
|
$$y=\begin{cases}2x-8 & (x>3) \\ x^{2} & (x\leqslant 3)\end{cases}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f65b400a-0cc1-42c5-b127-9a8dc8f0f5bd.png"
}
] |
cb5e52da-1106-4a6c-8add-1507ff4fa2ca
|
<image>As shown in the figure, $$AC$$ intersects $$BD$$ at point $$O$$, and $$AB=CD$$. Please add a condition ___ to make $$\triangle ABO\cong \triangle CDO$$.
|
$$\angle A=\angle C$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cb5e52da-1106-4a6c-8add-1507ff4fa2ca.png"
}
] |
26632821-7abc-4881-bd22-e9fb1b2a6428
|
<image>As shown in the figure, this is a movable rhombus clothes hanger with a side length of 13 cm, made based on the instability of a quadrilateral. To make the distance between points A and C, the two hooks, 10 cm, the fixed distance between points A and E is ______.
|
24cm
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/26632821-7abc-4881-bd22-e9fb1b2a6428.png"
}
] |
47b1a3ce-b51c-40f9-9e03-4663be09c6ad
|
<image>As shown in the figure, $$F_{1}$$ and $$F_{2}$$ are the two foci of the hyperbola $$\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1(a > 0,b > 0)$$, and $$A$$ and $$B$$ are the two intersection points of the circle centered at $$O$$ with radius $$|OF_{1}|$$ and the left branch of the hyperbola. Given that $$\triangle F_{2}AB$$ is an equilateral triangle, the eccentricity of the hyperbola is ___.
|
$$\sqrt{3}+1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/47b1a3ce-b51c-40f9-9e03-4663be09c6ad.png"
}
] |
c80b4c0b-a043-445a-8e9b-0f31a410c812
|
<image>As shown in the figure, in quadrilateral $$ABCD$$, points $$E$$ and $$F$$ are on the diagonal $$AC$$, and $$BE \parallel DF$$. Identify a pair of congruent triangles: ___.
|
$$\triangle ADF \cong \triangle BEC$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c80b4c0b-a043-445a-8e9b-0f31a410c812.png"
}
] |
a296e16a-fc5e-45ff-8821-d5b6cac11598
|
<image>As shown in the figure, in △ABC, AH⊥BC at point H, ∠C＝35°, and AB＋BH＝HC, then the measure of ∠B is ______ .
|
70°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a296e16a-fc5e-45ff-8821-d5b6cac11598.png"
}
] |
c9608fb7-2a51-4995-900a-8b9d37f41e13
|
<image>Arrange positive integers as shown in the figure: the number in the $$i$$th row and $$j$$th column is denoted as $$a^{j}_{i}$$, then the number $$2015$$ in the table should be denoted as ___.
|
$$a^{79}_{45}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c9608fb7-2a51-4995-900a-8b9d37f41e13.png"
}
] |
4ad9d563-5e42-446e-b595-7514356500f7
|
<image>Given the probability distribution of the random variable $$X$$ as follows: then the constant $$c=$$ ___.
|
$$\dfrac{1}{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4ad9d563-5e42-446e-b595-7514356500f7.png"
}
] |
c4ed651c-f3a8-4ac6-a50a-0aa9e5e1242a
|
<image>As shown in the figure, a square is formed by two identical rectangles, each with a width of $$1m$$. What is the area of the shaded part in $$cm^{2}$$?
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c4ed651c-f3a8-4ac6-a50a-0aa9e5e1242a.png"
}
] |
919944fe-e547-4c53-ae64-7e8392791798
|
<image>As shown in the figure, line $$MN$$ is tangent to circle $$\odot O$$ at point $$M$$, $$ME=EF$$, and $$EF \parallel MN$$. Then $$\cos E=$$ ___.
|
$$\dfrac{1}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/919944fe-e547-4c53-ae64-7e8392791798.png"
}
] |
d21ab730-e44b-40c3-a451-d2153d00d04f
|
<image>As shown in the figure, point $$E$$ is on $$CD$$, and $$Rt \triangle ACD \cong Rt \triangle EBC$$. Which of the following conclusions are correct: 1. $$AC = BC$$, 2. $$AD \parallel BE$$, 3. $$\angle ACB = 90^{°}$$, 4. $$AD + DE = BE$$. The number of correct conclusions is ___ .
|
$$1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d21ab730-e44b-40c3-a451-d2153d00d04f.png"
}
] |
daa81191-9d23-4064-8ca1-ad76cdf54686
|
<image>As shown in the figure, a segment of a parabola: $$y=-x(x-2)$$ $$\left(0\leqslant x\leqslant 2\right)$$ is denoted as $$C_1$$, which intersects the $$x$$-axis at points $$O$$ and $$A_1$$. By rotating $$C_1$$ around $$A_1$$ by $$180^\circ$$, we obtain $$C_2$$, which intersects the $$x$$-axis at $$A_2$$. By rotating $$C_2$$ around $$A_2$$ by $$180^\circ$$, we obtain $$C_3$$, which intersects the $$x$$-axis at $$A_3$$; and so on, until we get $$C_6$$. If point $$P\left(11,m\right)$$ lies on the 6th segment of the parabola $$C_6$$, then $$m=$$ ___.
|
$$-1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/daa81191-9d23-4064-8ca1-ad76cdf54686.png"
}
] |
4f20f9c0-c5ad-4f2d-9f3b-dc250a841c48
|
<image>As shown in the figure, $$F_{1}$$ and $$F_{2}$$ are the left and right foci of the ellipse $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\left ( a>b>0\right ) $$. Point $$P$$ is on the ellipse, and $$\triangle POF_{2}$$ is an equilateral triangle with an area of $$\sqrt{3}$$. Then $$a^{2}=$$ ___.
|
$$4+2\sqrt{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4f20f9c0-c5ad-4f2d-9f3b-dc250a841c48.png"
}
] |
1b8667f6-b311-40de-a8a3-9d48dcb601dc
|
<image>As shown in the figure, in $$\triangle ABC$$, $$AB=AC=2$$, $$BC=2\sqrt{3}$$, point $$D$$ is on $$BC$$, and $$\angle ADC=45^{\circ}$$, then the length of $$AD$$ is ___.
|
$$\sqrt{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1b8667f6-b311-40de-a8a3-9d48dcb601dc.png"
}
] |
2608ee21-3ce9-48c2-a4f2-d7d0c46d8d40
|
<image>As shown in the figure, $$\text{Rt}\triangle ABC\cong \text{Rt} \triangle DCB$$, the two hypotenuses intersect at point $$O$$. If $$AC=3$$, then the length of $$OD$$ is ___.
|
$$1.5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2608ee21-3ce9-48c2-a4f2-d7d0c46d8d40.png"
}
] |
3d5d2e49-13fd-44b3-bba2-8ccf107e2560
|
<image>As shown in the figure, the function $$f(x) = A \sin (\omega x + \varphi)$$ (where $$A > 0$$, $$\omega > 0$$, and $$|\varphi| \leqslant \dfrac{\pi}{2}$$) intersects the coordinate axes at three points $$P$$, $$Q$$, and $$R$$, satisfying $$P(1,0)$$, and $$M(2,-2)$$ is the midpoint of segment $$QR$$. The value of $$A$$ is ___.
|
$$\dfrac{8\sqrt{3}}{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3d5d2e49-13fd-44b3-bba2-8ccf107e2560.png"
}
] |
9562c52c-f447-4a5a-9498-485ca3d5bf95
|
<image>Execute the program flowchart as shown. If the inputs A and S are 0 and 1 respectively, then the output S = .
|
36
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9562c52c-f447-4a5a-9498-485ca3d5bf95.png"
}
] |
40c04eca-2d50-477f-adf5-19f07f4fad58
|
<image>Given the function $f(x)=\sin (\omega x+\varphi )(\omega > 0,0 < \varphi < \pi )$ with part of its graph shown in the figure, the value of $\varphi$ is.
|
$\frac{5\pi }{6}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/40c04eca-2d50-477f-adf5-19f07f4fad58.png"
}
] |
25d5b6d0-8739-40e3-ba27-660aef4f4085
|
<image>As shown in the figure, in the right triangle $\Delta ABC$, $\angle B={{90}^{\circ }}$, $AB=5$, $BC=12$. The triangle $\Delta ABC$ is rotated counterclockwise around point $A$ to form $\Delta ADE$, such that point $D$ lies on $AC$. Find the value of $\tan \angle ECD$.
|
$\frac{3}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/25d5b6d0-8739-40e3-ba27-660aef4f4085.png"
}
] |
8f9a4b19-5ab4-4707-89ca-86cc18c7d63a
|
<image>As shown in the figure, point $D$ is a moving point on side $BC$ of equilateral $\vartriangle ABC$. Connecting $AD$, the ray $DA$ is rotated $60^\circ$ clockwise around point $D$ and intersects $AB$ at point $E$. If $AB=4$, then the minimum value of $AE$ is .
|
$3$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8f9a4b19-5ab4-4707-89ca-86cc18c7d63a.png"
}
] |
bbe57f87-1665-48a9-a337-4a23a02e2f97
|
<image>As shown in the figure, a geometric solid is formed by combining two circular cones with the same base. If the base area is $9\pi$, and the heights of the smaller and larger cones are 4 and 6 respectively, then the surface area of the solid is.
|
$\left( 15+9\sqrt{5} \right)\pi $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bbe57f87-1665-48a9-a337-4a23a02e2f97.png"
}
] |
4b3211ca-fe50-46b3-9699-74e6f9adb480
|
<image>In the circle ⊙O, $\overset\frown{AB}=\overset\frown{AC}$, ∠A=30°, then ∠B=°.
|
75．
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4b3211ca-fe50-46b3-9699-74e6f9adb480.png"
}
] |
f8a1280f-3955-437d-8bd9-b9cd60ea6dfb
|
<image>As shown in the figure, quadrilateral $ABCD$ is a rectangle, point $E$ lies on the extension of line segment $CB$, and line $DE$ intersects $AB$ at point $F$. Given that $\angle AED = 2\angle CED$, point $G$ is the midpoint of $DF$, if $BE = 1$ and $DF = 7$, then the length of $AB$ is.
|
$\frac{3}{2}\sqrt{5}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f8a1280f-3955-437d-8bd9-b9cd60ea6dfb.png"
}
] |
254a1430-71cf-4353-b324-4ffd857c75bc
|
<image>In the right triangle ABC, ∠ACB = 90°, D, E, and F are the midpoints of AB, BC, and CA, respectively. If CD = 6 cm, then EF = cm.
|
6
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/254a1430-71cf-4353-b324-4ffd857c75bc.png"
}
] |
3e165e23-d7b1-4b0f-a317-6a530f90c0c3
|
<image>The following two figures are the views of an object made up of several identical small rectangular prisms. How many small rectangular prisms can be used to form this object at most?
|
$5$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3e165e23-d7b1-4b0f-a317-6a530f90c0c3.png"
}
] |
e0f435d9-e27d-4a30-a4d9-62577fd1c998
|
<image>As shown in the bar chart, it represents the age structure of a middle school basketball interest group at a certain school. The youngest age in the group is 13 years old, and the oldest is 17 years old. According to the data provided by the chart, the median age of the group members is .
|
15.5
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e0f435d9-e27d-4a30-a4d9-62577fd1c998.png"
}
] |
8ef12a1d-9266-44b7-8259-264d73cca532
|
<image>As shown in the figure, in the Cartesian coordinate system, the vertex A of the rhombus OABC is on the positive x-axis, and the coordinates of vertex C are (4,3). Point D is a point on the parabola y = -x^2 + 6x and is above the x-axis. The maximum value of the area of triangle BCD is ______.
|
15
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8ef12a1d-9266-44b7-8259-264d73cca532.png"
}
] |
2ad107a0-ee9a-4d22-8699-08b995e6c572
|
<image>Dad walks from point A to point B, and the son walks from point B to point A. They start at the same time. The relationship between the distance between them and the time the son has walked is shown in the figure. According to the graph, Dad walks ______ meters more per minute than the son.
|
30
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2ad107a0-ee9a-4d22-8699-08b995e6c572.png"
}
] |
166ec4ce-c299-45d5-b4e1-4ec526ff0e5a
|
<image>During the opening ceremony of the school sports meet, a flag-raising ceremony was held. The flagpole is directly in front of a certain row of the stands, which have a slope of $$15^{\circ}$$. The angles of elevation to the top of the flagpole from the first and last rows of this row are $$60^{\circ}$$ and $$30^{\circ}$$, respectively. The distance between the first and last rows is $$10\sqrt6$$ meters (as shown in the figure). The bottom of the flagpole is at the same level as the first row. If the national anthem lasts for $$50$$ seconds, the flag raiser should raise the flag at a uniform speed of ______ (meters/second).
|
$$0.6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/166ec4ce-c299-45d5-b4e1-4ec526ff0e5a.png"
}
] |
526f5190-2a19-4d7f-9c87-c6c68bed69ea
|
<image>A school has a total of 400 students who participated in a mathematics competition. The frequency distribution histogram of the competition scores is shown in the figure. The number of students who scored 80 points or above in this competition is.
|
160
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/526f5190-2a19-4d7f-9c87-c6c68bed69ea.png"
}
] |
6f8e1abb-17d9-4493-8ada-212f18fef38c
|
<image>As shown in the figure, given $\Delta ABC \cong \Delta ADE$, if $AB=7, AC=3$, then the value of $BE$ is.
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6f8e1abb-17d9-4493-8ada-212f18fef38c.png"
}
] |
665d6a73-40be-4d57-8002-0e1dab9e42b8
|
<image>As shown in the figure, the side lengths of two squares are a and b, respectively. If a + b = 20 and ab = 30, then the area of the shaded part is.
|
155
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/665d6a73-40be-4d57-8002-0e1dab9e42b8.png"
}
] |
2a184afb-5170-4bbe-a25e-9f0cde3a7d5e
|
<image>As shown in the figure, point P is inside ∠AOB, PE⊥OA, PF⊥OB, with the feet of the perpendiculars being E and F, respectively. If PE＝PF, and ∠OPF＝72°, then the degree measure of ∠AOB is.
|
36°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2a184afb-5170-4bbe-a25e-9f0cde3a7d5e.png"
}
] |
7343a69b-6641-4266-94e5-54fed31efdd2
|
<image>As shown in the figure, parallelogram ABCD is rotated 30° counterclockwise around point A, resulting in parallelogram AB'C'D' (point B corresponds to point B', point C corresponds to point C', and point D corresponds to point D'). Point B' exactly lands on side BC, then ∠C = ______ degrees.
|
105°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7343a69b-6641-4266-94e5-54fed31efdd2.png"
}
] |
7a72a2fd-183a-485b-8ebb-d2ab63aba59a
|
<image>Execute the program shown in the figure, if the input is $$m=30$$, $$n=18$$, then the output result is ___.
|
$$6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7a72a2fd-183a-485b-8ebb-d2ab63aba59a.png"
}
] |
c202a1cd-f5fd-4b4a-bf6f-4b9b39f06894
|
<image>As shown in the figure, in $$\triangle ABC$$, it is known that $$AB=4$$, $$AC=6$$, and $$\angle BAC=60^{\circ}$$, points $$D$$ and $$E$$ are on sides $$AB$$ and $$AC$$ respectively, and $$\overrightarrow{AB}=2\overrightarrow{AD}$$, $$\overrightarrow{AC}=3\overrightarrow{AE}$$. Point $$F$$ is the midpoint of $$DE$$. Then the value of $$\overrightarrow{BF}\cdot \overrightarrow{DE}$$ is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c202a1cd-f5fd-4b4a-bf6f-4b9b39f06894.png"
}
] |
a081b0ae-352d-4337-be30-52efbc9941f4
|
<image>As shown in the figure, $$\odot O$$ is the circumcircle of $$\triangle ABC$$, $$AD$$ is the diameter of $$\odot O$$, and $$CD$$ is connected. Given $$\angle B=70^{\circ}$$, then $$\angle DAC=$$ _____.
|
$$20^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a081b0ae-352d-4337-be30-52efbc9941f4.png"
}
] |
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