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As shown in the figure, in the rhombus ABCD, $AC=8$, $BD=6$. Let $E$ be a moving point on side $CD$. Draw $EF\perp OC$ at point $F$ and $EG\perp OD$ at point $G$, and connect $FG$. What is the minimum value of $FG$?
|
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[
"images/ddb19b7d-6527-404d-9e7a-41c73fac4bb8-0.png"
] | null | 1,837
|
As shown in the diagram, in the rhombus $ABCD$, $\angle ABC=30^\circ$. Connect $AC$. Follow the steps below to draw: Take $C$ and $B$ as centers, and with the length of $BC$ as the radius, draw arcs. The arcs intersect at point $P$. Connect $BP$ and $CP$. What is the measure of $\angle ACP$?
|
[
"\\boxed{15}"
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|
[
"images/aed89320-4af7-4da4-a380-07d10b8098d1-0.png"
] | null | 1,838
|
As shown in the figure, points $E$, $F$, $G$, and $H$ are the midpoints of the sides of square $ABCD$, and lines $BE$, $DG$, $CF$, $AH$ are connected. If $AB=10$, what is the area of quadrilateral $MNPQ$?
|
[
"\\boxed{20}"
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[
"images/a60df5cf-d986-4bd5-9ebf-fcea8dfa876e-0.png"
] | null | 1,839
|
As shown in the figure, in rectangle $ABCD$, if $AB=7$ and $\angle DBC=30^\circ$, what is the length of $AC$?
|
[
"\\boxed{14}"
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d8b1b588-0d30-4d97-b27a-e3eae82e7f6b
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[
"images/d8b1b588-0d30-4d97-b27a-e3eae82e7f6b-0.png"
] | null | 1,840
|
In the figure, within $\triangle ABC$, $\angle ACB=90^\circ$, CD is the median. Extend CB to point E such that BE=BC; connect DE, let F be the midpoint of DE, and connect BF. If AC=8, BC=6, what is the length of BF?
|
[
"\\boxed{\\frac{5}{2}}"
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|
[
"images/e99f1cd9-0456-4ca7-b611-430d18a9abf8-0.png"
] | null | 1,841
|
As shown in the figure, given point $E$ is on the edge $AB$ of the rectangular sheet $ABCD$. The sheet is folded along $DE$ such that the corresponding point of $A$, denoted as $A'$, falls exactly on the line segment $CE$. If $AD=3$ and $AE=1$, then what is the length of $CD$?
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[
"\\boxed{5}"
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"images/5f59ddd4-63f7-476e-ade1-ef39285f06ca-0.png"
] | null | 1,842
|
As shown in the figure, points $D$ and $E$ are on the sides of $\triangle ABC$. The angle bisector of $\angle ABC$ is perpendicular to $AE$ with the foot of the perpendicular being $Q$, and the angle bisector of $\angle ACB$ is perpendicular to $AD$ with the foot of the perpendicular being $P$. If $BC=12$ and $PQ=3$, what is the perimeter of $\triangle ABC$?
|
[
"\\boxed{30}"
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[
"images/81d4510b-21b4-490a-87da-97c284df118c-0.png"
] | null | 1,843
|
As shown in the diagram, in $\triangle ABC$, $AB=AC$, $\angle B=25^\circ$, and point D is on side $BC$. Fold $\triangle ACD$ along $AD$ to get $\triangle ADE$. If $AD=DE$, then what is the measure of $\angle AFB$?
|
[
"\\boxed{75}"
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[
"images/0f6198f9-c9b9-44cf-981f-218b51aaaf95-0.png"
] | null | 1,844
|
As shown in the figure, in rectangle $ABCD$, $AB=8\,\text{cm}$, $BC=10\,\text{cm}$. A point $E$ is taken on side $CD$, and $\triangle ADE$ is folded such that point $D$ aligns exactly with point $F$ on side $BC$. What is the length of $CE$ in $\text{cm}$?
|
[
"\\boxed{3}"
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[
"images/5b971fdd-5d2b-42a4-9784-2f12d30cbc69-0.png"
] | null | 1,845
|
As shown in the figure, in rectangle $ABCD$, $E$ and $F$ are the midpoints of the sides $CD$ and $AD$ respectively, $CF$ intersects $EA$ and $EB$ at points $M$ and $N$ respectively. Given $AB=8$ and $BC=12$, what is the length of $MN$?
|
[
"\\boxed{\\frac{8}{3}}"
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[
"images/0a6d0a4f-2624-4d2f-bf37-90286fc7e88b-0.png"
] | null | 1,846
|
As shown in the diagram, point $D$ is on side $AB$ of $\triangle ABC$, and point $E$ is the midpoint of $CD$. Connect $AE$ and $BE$. If the area of $\triangle ABC$ is 8, what is the area of the shaded region?
|
[
"\\boxed{4}"
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[
"images/c5dfb057-0ac6-448c-bc77-1f338546edb7-0.png"
] | null | 1,847
|
\textit{As shown in the figure, in an equilateral $\triangle ABC$, $BD$ is the bisector of $\angle ABC$, point $E$ is the midpoint of $BC$, and point $P$ is a movable point on $BD$. Lines $PE$ and $PC$ are drawn. When the value of $PE + PC$ is at its minimum, what is the degree measure of $\angle EPC$?}
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[
"\\boxed{60}"
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[
"images/7e979ef7-38a8-42f9-bc63-bc692ea1ee31-0.png"
] | null | 1,848
|
As shown in the figure, in $\triangle ABC$, $\angle C=90^\circ$, the side $DE$ of rectangle $DEFG$ is on side $AB$, and the vertices $F$ and $G$ are on sides $BC$ and $AC$, respectively. If the areas of $\triangle BEF$, $\triangle ADG$, and $\triangle CFG$ are respectively 1, 2, and 3, then what is the area of rectangle $DEFG$?
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[
"\\boxed{6}"
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[
"images/c73d7b2e-b1d4-4a82-ad05-02e60b6f6d47-0.png"
] | null | 1,849
|
As shown in the figure, square $ABCD$ has a side length of $4$, point $M$ is on $AB$ with $AM=1$, and $N$ is a moving point on $BD$. What is the minimum value of $AN+MN$?
|
[
"\\boxed{5}"
] |
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|
[
"images/d8b6c671-7374-4f75-8c66-f1ada024e698-0.png"
] | null | 1,850
|
As shown in the figure, in $\triangle ABC$, point $D$ is on side $AB$, $E$ is the midpoint of side $AC$, $CF\parallel AB$, and the extension lines of $CF$ and $DE$ intersect at point $F$. If $AB=4$ and $CF=3$, what is the length of $BD$?
|
[
"\\boxed{1}"
] |
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|
[
"images/9e5765d8-0b81-4295-90e7-db81faedaaf2-0.png"
] | null | 1,851
|
As shown in the figure, in $\triangle ABC$, $\angle C=37^\circ$, the perpendicular bisector of side $BC$ intersects $AC$ and $BC$ at points $D$ and $E$, respectively, and $AB=CD$, then what is the degree measure of $\angle A$?
|
[
"\\boxed{74}"
] |
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|
[
"images/474ae0a4-6e99-4bd1-9b4d-b66cc074fcfd-0.png"
] | null | 1,852
|
As shown in the figure, $\angle AOE = \angle BOE = 15^\circ$, $EF \parallel OB$, $EC \perp OB$. If $EC = 2$, what is the length of $EF$?
|
[
"\\boxed{4}"
] |
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|
[
"images/94943140-afea-46d0-9e37-36baa72a791b-0.png"
] | null | 1,853
|
As shown in the figure, it is known that in $\triangle ABC$, $\angle C=90^\circ$, $BC=3$, and $AC=5$. The triangle is folded along $DE$ such that point $A$ coincides with point $B$. What is the length of $AE$?
|
[
"\\boxed{3.4}"
] |
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|
[
"images/892be125-fbaf-444e-8989-9ad9f5f7b77a-0.png"
] | null | 1,854
|
As shown in the figure, the line $y=-2x+2$ intersects the $x$-axis and the $y$-axis at points $A$ and $B$, respectively, and the ray $AP\perp AB$ at point $A$. If point $C$ is a moving point on ray $AP$, and point $D$ is a moving point on the $x$-axis, and the triangle with vertices $C$, $D$, $A$ is congruent to $\triangle AOB$, then what is the length of $OD$?
|
[
"\\boxed{3}"
] |
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|
[
"images/a9f390f8-110b-4fe2-a137-269cc0a68b58-0.png"
] | null | 1,855
|
As shown in the figure, the line $y=kx-3$ intersects the x-axis and y-axis at points B and A, respectively, with $OB=\frac{1}{3}OA$. Point C is a point on line AB, located in the second quadrant. When the area of $\triangle OBC$ is 3, what are the coordinates of point C?
|
[
"\\boxed{\\left(-3,6\\right)}"
] |
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|
[
"images/408ea28a-9b98-4f5d-90cf-8bb278cb6f76-0.png"
] | null | 1,856
|
As shown in the figure, in the Cartesian coordinate system, the graph of the linear function $y=kx+b$ passes through the vertices A and C of the square OABC. Given that the coordinates of point A are $\left(1, -3\right)$, what is the value of $b$?
|
[
"\\boxed{-5}"
] |
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|
[
"images/025f9b48-a720-47a3-a525-8023b133f51c-0.png"
] | null | 1,857
|
As shown in the figure, in $\triangle ABC$, we have $AB=AC$, $AD=BC=8$. Points $E$ and $F$ are on the median $AD$. What is the area of the shaded region?
|
[
"\\boxed{16}"
] |
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[
"images/269a45dd-f7cf-4371-81ef-69005180256a-0.png"
] | null | 1,858
|
As shown in the figure, in $\triangle ABC$, $AB=AC$, $\angle BAC=90^\circ$, point $D$ is on segment $BC$, $\angle EDB=\frac{1}{2}\angle ACB$, $BE\perp DE$, and $DE$ intersects $AB$ at point $F$. If $BE=4$, what is the length of $DF$?
|
[
"\\boxed{8}"
] |
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|
[
"images/37ace0e0-8197-4605-a748-f4cd7c3f2939-0.png"
] | null | 1,859
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, $AC=BC$, the coordinates of point $A$ are $(-7,3)$, and the coordinates of point $C$ are $(-2,0)$. What are the coordinates of point $B$?
|
[
"\\boxed{(1, 5)}"
] |
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|
[
"images/7f8324a5-ae23-4f03-8013-ea2b534f4eb9-0.png"
] | null | 1,860
|
As shown in the figure, two congruent right triangles overlap each other. One of the triangles is translated along the direction from point B to C to the position of $\triangle DEF$, where $AB=10$, $DO=4$, and the translation distance is $6$. What is the area of the shaded region?
|
[
"\\boxed{48}"
] |
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|
[
"images/00353204-1b61-43cc-a64b-e5983d27cf59-0.png"
] | null | 1,861
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, isosceles right triangles $S_1$ and $S_2$ are constructed on hypotenuses $AC$ and $BC$, respectively, and a square $S$ is constructed on side $AB$. If the sum of the areas of $S_1$ and $S_2$ equals $9$, what is the length of the side of square $S$?
|
[
"\\boxed{6}"
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|
[
"images/11384fa8-52f4-4b26-93c1-04a92d437db9-0.png"
] | null | 1,862
|
As shown in the figure, the quadrilateral $ABCO$ is a rectangle. The coordinates of point $A$ are $(4, 0)$, and the coordinates of point $C$ are $(0, 8)$. When the rectangle $ABCO$ is folded along $OB$, point $A$ falls on point $D$. What are the coordinates of point $D$?
|
[
"\\boxed{\\left(-\\frac{12}{5}, \\frac{16}{5}\\right)}"
] |
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[
"images/acc96e9d-502c-4ef1-a987-8758e771da11-0.png"
] | null | 1,863
|
In the figure, inside $\triangle ABC$, the median from $AB$ intersects side $AC$ at point $E$, with $BE=5$ and $CE=3$. What is the length of $AC$?
|
[
"\\boxed{8}"
] |
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|
[
"images/af47a92d-6eee-4b9e-85e1-e096da65b4f0-0.png"
] | null | 1,864
|
As shown in the figure, in $ \triangle ABC$, D is a point on $BC$, $AD=BD$, $\angle B=40^\circ$, and $\triangle ADE$ is symmetrical to $\triangle ADB$ with respect to $AD$. What is the measure of $\angle CDE$?
|
[
"\\boxed{20}"
] |
6694d2fe-c446-4f4b-b2c0-2ea96202e82b
|
[
"images/6694d2fe-c446-4f4b-b2c0-2ea96202e82b-0.png"
] | null | 1,865
|
As shown in the figure, it is known that point $A(0, 2)$, $B(4, 0)$, point $C$ is on the $x$ axis, $CD\perp x$ axis, intersecting line segment $AB$ at point $D$, and point $D$ does not coincide with points $A$, $B$. Fold $\triangle ABO$ along $CD$, making point $B$ fall at point $E$ on the $x$ axis. Let the abscissa of point $C$ be $x$. Then, when $\triangle ADE$ is a right-angled triangle, what is the value of $x$?
|
[
"\\boxed{1.5}"
] |
9920b10a-18d6-4a09-8343-0840af05bcc7
|
[
"images/9920b10a-18d6-4a09-8343-0840af05bcc7-0.png"
] | null | 1,866
|
As shown in the figure, in quadrilateral \(ABCD\), \(AD \parallel BC\), \(\angle BAD=90^\circ\), and diagonals \(BD \perp DC\), \(AD=4\), \(BC=9\), what is the length of \(BD\)?
|
[
"\\boxed{6}"
] |
9ec56e32-485c-4563-8802-0cd31f122fe9
|
[
"images/9ec56e32-485c-4563-8802-0cd31f122fe9-0.png"
] | null | 1,867
|
As shown in the figure, within circle $\odot O$, diameter $AB \perp CD$, and $\angle A=26^\circ$, what is the degree measure of the inscribed angle subtended by chord $AC$?
|
[
"\\boxed{64}"
] |
0ec277fd-ba00-449c-b54d-15409b06d70a
|
[
"images/0ec277fd-ba00-449c-b54d-15409b06d70a-0.png"
] | null | 1,868
|
As shown in the figure, in $\triangle ABC$, $\angle BAC=120^\circ$, $AB=AC=6$, $D$ is a moving point on side $AB$ (not coinciding with point $B$). Connect $CD$, and rotate line segment $CD$ $90^\circ$ counterclockwise around point $D$ to obtain $DE$. Connect $BE$. What is the maximum value of $S_{\triangle BDE}$?
|
[
"\\boxed{\\frac{81}{8}}"
] |
6916675e-8597-4d22-acf2-3ef03548c335
|
[
"images/6916675e-8597-4d22-acf2-3ef03548c335-0.png"
] | null | 1,869
|
As shown in the figure, in the Cartesian coordinate system, the perpendiculars to the x-axis and y-axis through the point $M(-3, 2)$ intersect the graph of the inverse proportion function $y=\frac{4}{x}$ at points A and B, respectively. What is the area of the quadrilateral MAOB?
|
[
"\\boxed{10}"
] |
1023d48f-b91e-4273-bd92-8d2872f0023b
|
[
"images/1023d48f-b91e-4273-bd92-8d2872f0023b-0.png"
] | null | 1,870
|
As shown in the figure, in the rhombus $ABCD$, $\angle C = 108^\circ$, the perpendicular bisector of $AD$ intersects the diagonal $BD$ at point $P$, and the foot of the perpendicular is $E$. Connecting $AP$, what is the measure of $\angle APB$?
|
[
"\\boxed{72}"
] |
63247700-8c3e-48f8-aaaa-0c545caab8cd
|
[
"images/63247700-8c3e-48f8-aaaa-0c545caab8cd-0.png"
] | null | 1,871
|
As shown in the figure, in $\triangle ABC$, D and E are the midpoints of AB and AC, respectively. Extend DE to the point F such that EF=DE. Given AB=10 and BC=8, what is the perimeter of the quadrilateral BCFD?
|
[
"\\boxed{26}"
] |
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|
[
"images/60400d0a-ff83-4330-b35e-6b3f78095860-0.png"
] | null | 1,872
|
As shown in the figure, $AB$ is the tangent to circle $O$ at point $A$, $OB$ intersects circle $O$ at point $C$, point $D$ is on circle $O$, and $AD$, $CD$, $OA$ are connected. If $\angle ADC = 25^\circ$, what is the degree measure of $\angle B$?
|
[
"\\boxed{40}"
] |
4727f76a-928f-4452-9154-6e0547cb2105
|
[
"images/4727f76a-928f-4452-9154-6e0547cb2105-0.png"
] | null | 1,873
|
In square $ABCD$, where $AB = 2$, let $E$ be the midpoint of $AB$. If $\triangle ADE$ is folded along $DE$ to form $\triangle FDE$, and $FH \perp BC$ with the foot of the perpendicular being $H$, then what is the length of $FH$?
|
[
"\\boxed{\\frac{2}{5}}"
] |
102d9cd4-1a5a-4b7c-95c0-e9a2084a9772
|
[
"images/102d9cd4-1a5a-4b7c-95c0-e9a2084a9772-0.png"
] | null | 1,874
|
As shown in the figure, given the quadratic function $y=-x^2+2x+m$ with its partial graph illustrated, what is the solution set of the quadratic inequality $-x^2+2x+m>0$ with respect to $x$?
|
[
"\\boxed{-1<x<3}"
] |
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|
[
"images/56f8a48d-2ce4-446e-9726-dd07ce2e1f73-0.png"
] | null | 1,875
|
As shown in the figure, points B and C are on the line segment AD, and the ratio of AB:BC:CD is 2:3:4. Let M be the midpoint of line segment AC, and let N be the midpoint of line segment CD, with MN=9. Find the length of BD.
|
[
"\\boxed{14}"
] |
dde57a2b-6530-4f7e-9b38-e9dbf6b05bab
|
[
"images/dde57a2b-6530-4f7e-9b38-e9dbf6b05bab-0.png"
] | null | 1,876
|
As shown in the figure, $AD$ and $BE$ are the altitude and angle bisector of $\triangle ABC$, respectively, with $\angle BAC=86^\circ$ and $\angle C=58^\circ$. Find the measure of $\angle AOB$?
|
[
"\\boxed{108}"
] |
a5c217f9-8473-4343-adce-f5f47f8e5444
|
[
"images/a5c217f9-8473-4343-adce-f5f47f8e5444-0.png"
] | null | 1,877
|
As shown in the figure, M is the midpoint of CD, EM$\perp$CD. If CD$=4$ and EM$=6$, what is the radius of the circle containing arc $\overset{\frown}{CED}$?
|
[
"\\boxed{\\frac{10}{3}}"
] |
30572859-0714-476b-a7af-5bca8f750baa
|
[
"images/30572859-0714-476b-a7af-5bca8f750baa-0.png"
] | null | 1,878
|
As shown in the figure, in $\triangle ABC$, $AB=AC$, $\angle A=50^\circ$, and the perpendicular bisector of $AB$, $DE$, intersects $AC$ and $AB$ at points $D$ and $E$, respectively. What is the degree measure of $\angle CBD$?
|
[
"\\boxed{15}"
] |
29e5d27e-6822-48da-9c6f-6317ed22d1f4
|
[
"images/29e5d27e-6822-48da-9c6f-6317ed22d1f4-0.png"
] | null | 1,879
|
As shown in the figure, in $\triangle ABC$, $AB=AC$, and the bisectors of $\angle BAC$ and $\angle ACB$ intersect at point $D$. Given that $\angle ADC=130^\circ$, find the measure of $\angle BAC$?
|
[
"\\boxed{20}"
] |
776cf38b-187e-48e0-8a38-0f2a7d56a0a4
|
[
"images/776cf38b-187e-48e0-8a38-0f2a7d56a0a4-0.png"
] | null | 1,880
|
As shown in the figure, within $\triangle ABC$, $AD$ is the height of $\triangle ABC$, $AE$ and $BF$ are the angle bisectors of $\triangle ABC$, and $AE$ intersects $BF$ at point $O$. Given that $\angle BOA = 125^\circ$, what is the measure of $\angle DAC$ in degrees?
|
[
"\\boxed{20}"
] |
a133c783-fe51-4449-b167-1cc4007528a5
|
[
"images/a133c783-fe51-4449-b167-1cc4007528a5-0.png"
] | null | 1,881
|
In the rhombus $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$, with $AC=8$, $BD=6$, and $OE\perp BC$ with the foot of the perpendicular being point $E$. Find the length of $OE$.
|
[
"\\boxed{\\frac{12}{5}}"
] |
a57e56e1-df72-4cec-b503-94aad214e527
|
[
"images/a57e56e1-df72-4cec-b503-94aad214e527-0.png"
] | null | 1,882
|
As shown in the figure, in $\triangle ABC$, $\angle ABC = 45^\circ$. $F$ is the intersection point of altitudes $AD$ and $BE$, $AC = \sqrt{5}$, $BD = 2$. What is the length of segment $DF$?
|
[
"\\boxed{1}"
] |
b4befa7e-bc5a-410b-ae7e-7f9d86588b56
|
[
"images/b4befa7e-bc5a-410b-ae7e-7f9d86588b56-0.png"
] | null | 1,883
|
As shown, it is known that in $\triangle ABC$, $AD$ is the median to $BC$, and point $E$ is on $AD$ with $\frac{DE}{AE} = \frac{1}{3}$. The ray $CE$ intersects $AB$ at point $F$. Find the value of $\frac{AF}{FB}$.
|
[
"\\boxed{\\frac{3}{2}}"
] |
83c2734a-8b27-4207-81a4-0907856a3ded
|
[
"images/83c2734a-8b27-4207-81a4-0907856a3ded-0.png"
] | null | 1,884
|
As shown in the figure, it is known that: $OA\perp OB$, $\angle BOC=50^\circ$, and $\angle AOD : \angle COD=4 : 7$. Draw the angle bisector OE of $\angle BOC$, and find the degree measure of $\angle DOE$.
|
[
"\\boxed{165}"
] |
b02cbc11-ee75-49aa-96f4-2a4a029bd1fa
|
[
"images/b02cbc11-ee75-49aa-96f4-2a4a029bd1fa-0.png"
] | null | 1,885
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, $\angle B=50^\circ$. CD is the altitude of $\triangle ABC$, and CE is the angle bisector of $\triangle ABC$. Find the measure of $\angle DCE$.
|
[
"\\boxed{5}"
] |
56e88717-4429-4267-87ad-eb01c6002cf2
|
[
"images/56e88717-4429-4267-87ad-eb01c6002cf2-0.png"
] | null | 1,886
|
As shown in the figure, the straight lines AB and CD intersect at point O, with $\angle 1: \angle 2 = 2:3$. Find the measure of $\angle BOD$ in degrees?
|
[
"\\boxed{72}"
] |
d38e3264-d43d-46db-b232-e914aa642359
|
[
"images/d38e3264-d43d-46db-b232-e914aa642359-0.png"
] | null | 1,887
|
As shown in the figure, OD is the bisector of $\angle BOC$, OE is the bisector of $\angle AOC$, and $\angle AOB : \angle BOC = 3 : 2$. If $\angle BOE = 13^\circ$, what is the degree measure of $\angle DOE$?
|
[
"\\boxed{39}"
] |
ce8d4386-908f-4d51-b1ee-fc6062cfc27f
|
[
"images/ce8d4386-908f-4d51-b1ee-fc6062cfc27f-0.png"
] | null | 1,888
|
In rectangle $ABCD$, $AB=4$, $BC=5$, P is a moving point on ray $BC$, draw $PE\perp AP$, $PE$ intersects ray $DC$ at point E, ray $AE$ intersects ray $BC$ at point F, let $BP=x$, $CF=y$. When $\sin\angle APB=\frac{4}{5}$, what is the length of $CE$?
|
[
"\\boxed{\\frac{3}{2}}"
] |
ed174167-24f4-41b3-829a-b3ac79d5ed22
|
[
"images/ed174167-24f4-41b3-829a-b3ac79d5ed22-0.png"
] | null | 1,889
|
As shown in the figure, it is known that the graph of the inverse proportion function \(y_{1}=\frac{k}{x}\) and the graph of the linear function \(y_{2}=x+b\) intersect at point \(A(1, 4)\) and point \(B(-4, n)\). Directly write out the range of values of the independent variable \(x\) when \(y_{2}>y_{1}\).
|
[
"\\boxed{x \\in (-4, 0) \\cup (1, +\\infty)}"
] |
020c675f-ea04-4a42-a233-b5b693d61206
|
[
"images/020c675f-ea04-4a42-a233-b5b693d61206-0.png"
] | null | 1,890
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, $AC=BC$, point $C(2, 0)$, point $B(0, 4)$, and the graph of the inverse proportion function $y=\frac{k}{x} (k\ne 0)$ passes through point A. The line $OA$ is translated upwards by $m$ units and then passes through the point $(3, n)$ on the graph of the inverse proportion function $y=\frac{k}{x} (k\ne 0)$. Find the value of $m$?
|
[
"\\boxed{1}"
] |
be1ef304-f5a3-4fee-9d00-28a1cacb0936
|
[
"images/be1ef304-f5a3-4fee-9d00-28a1cacb0936-0.png"
] | null | 1,891
|
As shown in the figure, the parabola $y=ax^2-a$ ($a>0$) intersects the x-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and intersects the y-axis at point $C$, with $OC=2OA$. Is there a point $P$ on the parabola such that the incenter of $\triangle PAC$ lies on the x-axis? If it exists, find the coordinates of point $P$; if not, please explain why.
|
[
"\\boxed{(2,6)}"
] |
2a473215-b62d-465d-807a-98f3eb326ec1
|
[
"images/2a473215-b62d-465d-807a-98f3eb326ec1-0.png"
] | null | 1,892
|
As shown in the figure, in the Cartesian coordinate system, the parabola $C_1\colon y=-\frac{1}{4}x^2+bx+c$ passes through the points $(-3,0)$ and $(3,3)$, and intersects the y-axis at point C. The parabola $C_1$ is first translated 3 units to the left and then 2 units downward to obtain the parabola $C_2$. The vertex of parabola $C_2$ is D. What is the area of $\triangle ODC$?
|
[
"\\boxed{\\frac{15}{4}}"
] |
b53f3bac-05bf-4bf9-b5b2-7ff183aeb810
|
[
"images/b53f3bac-05bf-4bf9-b5b2-7ff183aeb810-0.png"
] | null | 1,893
|
As shown in the figure, point \(A\) lies on the extension line of diameter \(CD\) of circle \(\odot O\), and point \(B\) is on circle \(\odot O\), with \(\angle A=\angle C=30^\circ\). Line \(AB\) is a tangent to circle \(\odot O\); if \(BC=2\sqrt{3}\), what is the length of \(AC\)?
|
[
"\\boxed{6}"
] |
7d0e693a-c63e-4f0c-8785-43cde98b39aa
|
[
"images/7d0e693a-c63e-4f0c-8785-43cde98b39aa-0.png"
] | null | 1,894
|
As shown in the figure, $AB$ is the diameter of $\odot O$, and $OC$ is the radius. Connect $AC$ and $BC$. Extend $OC$ to point $D$ such that $\angle CAD = \angle B$. Draw $DM\perp AD$ meeting the extension of $AC$ at point $M$. $AD$ is the tangent line of $\odot O$. If $AD=6$ and $\tan\angle CAD=\frac{1}{2}$, find the length of the radius of $\odot O$.
|
[
"\\boxed{\\frac{9}{2}}"
] |
c689eda5-cb43-4020-86e2-83176094da13
|
[
"images/c689eda5-cb43-4020-86e2-83176094da13-0.png"
] | null | 1,895
|
As shown in the figure, $AB$ is the diameter of $\odot O$, the chord $CD\perp AB$ at point $E$, and chord $AF$ intersects chord $CD$ at point $G$, with $AG=CG$. A perpendicular line to $BF$ through point $C$ intersects the extension of $BF$ at point $H$. $CH$ is the tangent of $\odot O$; if $FH=2$ and $BF=4$, what is the length of arc $CD$?
|
[
"\\boxed{\\frac{8}{3}\\pi}"
] |
99dda302-7d8f-4325-9c2f-82a7d7a678ac
|
[
"images/99dda302-7d8f-4325-9c2f-82a7d7a678ac-0.png"
] | null | 1,896
|
As shown in the diagram, in $\triangle ABC$, $\angle ACB=90^\circ$, point O is a point on side BC, a circle with center O and radius OB intersects side AB at point D, connect DC, and $DC=AC$. $DC$ is the tangent to $\odot O$; If the radius of $\odot O$ is 3, and $CD=4$, find the length of BC.
|
[
"\\boxed{8}"
] |
fb819e6b-57af-45a0-bbac-48a19d916519
|
[
"images/fb819e6b-57af-45a0-bbac-48a19d916519-0.png"
] | null | 1,897
|
As shown in the figure, with AB as the diameter, construct $\odot O$. Take a point C on $\odot O$, extend AB to point D, and connect DC, where $\angle DCB=\angle DAC$. Through point A, draw $AE\perp AD$ to meet the extension of DC at point E. CD is the tangent to $\odot O$; if $CD=4$ and $DB=2$, find the length of AE.
|
[
"\\boxed{6}"
] |
60d437ca-c5a5-4cd5-9b54-f142626196f0
|
[
"images/60d437ca-c5a5-4cd5-9b54-f142626196f0-0.png"
] | null | 1,898
|
As shown in the diagram, within quadrilateral $ABCD$, points $E$ and $F$ are on the diagonal $AC$ such that $AE=EF=FC$. Quadrilateral $DEBF$ is a parallelogram; if $\angle CDE=90^\circ$, $DC=8$, and $DE=6$, what is the perimeter of quadrilateral $DEBF$?
|
[
"\\boxed{22}"
] |
65f3c9ca-4bfc-48fe-a023-1f4094146b98
|
[
"images/65f3c9ca-4bfc-48fe-a023-1f4094146b98-0.png"
] | null | 1,899
|
As shown in the figure, square $ABEF$ and square $ACGD$ are constructed externally on sides $AB$ and $AC$ of $\triangle ABC$, respectively. Lines $BD$, $CF$, and $DF$ are drawn, with $CF$ intersecting $BD$ at $O$ and $AD$ at $H$. If $AB=2$ and $AC=4$, what is the value of $BC^2+DF^2$?
|
[
"\\boxed{40}"
] |
14a074a1-4d34-43c1-ad58-18eb72a1570b
|
[
"images/14a074a1-4d34-43c1-ad58-18eb72a1570b-0.png"
] | null | 1,900
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, $AC=3$, and $BC=4$. Squares $ABMN$, $ACDE$, and $BCFG$ are constructed on the sides of $\triangle ABC$ on the same side as $AB$. Point $M$ is on side $FG$, $MN$ intersects $CF$ at point $P$, and $AN$ intersects $CD$ at point $Q$. $\triangle ABQ \cong \triangle NAP$; what is the area of quadrilateral $CPNQ$?
|
[
"\\boxed{6}"
] |
7aaacc07-c284-41bf-a64b-ae369e3d083e
|
[
"images/7aaacc07-c284-41bf-a64b-ae369e3d083e-0.png"
] | null | 1,901
|
As shown in the figure, in $\triangle ABC$, $AB=CB$, $\angle ABC=90^\circ$, $F$ is a point on the extension of $AB$, and point $E$ is on $BC$, with $BE=BF$. $\triangle ABE \cong \triangle CBF$. If $\angle CAE = 20^\circ$, what is the measure of $\angle ACF$?
|
[
"\\boxed{70}"
] |
bdad6851-16e6-4a51-8bb0-8aeccb6ca83c
|
[
"images/bdad6851-16e6-4a51-8bb0-8aeccb6ca83c-0.png"
] | null | 1,902
|
As shown in the figure, in $\triangle ABC$, $D$ is a point on side $BC$, and $AB=AD=DC$. If $AC=BC$, what is the degree measure of $\angle C$?
|
[
"\\boxed{36}"
] |
04bf8d56-ed22-4dd6-8cc9-d7d62227a623
|
[
"images/04bf8d56-ed22-4dd6-8cc9-d7d62227a623-0.png"
] | null | 1,903
|
Given: As shown in the figure, in rectangle $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$. Lines parallel to $AC$ and $BD$ are drawn passing through points $B$ and $C$, respectively, and intersect at point $F$. Quadrilateral $BFCO$ is a rhombus. If $AB=3$ and $BC=4$, what is the perimeter of quadrilateral $BFCO$?
|
[
"\\boxed{10}"
] |
dbfdf4e1-dccf-48cf-9873-68fa650433b7
|
[
"images/dbfdf4e1-dccf-48cf-9873-68fa650433b7-0.png"
] | null | 1,904
|
As shown in the figure, the lines $EF$ and $CD$ intersect at the point $O$, with $\angle AOB = 90^\circ$, and $OC$ bisects $\angle AOF$. If $\angle AOE = 40^\circ$, find the degree measure of $\angle BOD$.
|
[
"\\boxed{20}"
] |
f3a01dff-2a20-4879-bbae-f27e7c85b818
|
[
"images/f3a01dff-2a20-4879-bbae-f27e7c85b818-0.png"
] | null | 1,905
|
Given that point $C$ is on line $AB$, and points $M$ and $N$ are the midpoints of $AC$ and $BC$, respectively. If point $C$ is on the extension of segment $AB$, and it satisfies $AC-BC=a$ cm, please draw a diagram according to the given conditions, and calculate the length of $MN$ (express the result in terms of $a$).
|
[
"\\boxed{\\frac{1}{2}a}"
] |
e7bbcabc-b85f-43bd-aae0-04afe870e975
|
[
"images/e7bbcabc-b85f-43bd-aae0-04afe870e975-0.png"
] | null | 1,906
|
Given that $\angle AOB = 90^\circ$, as shown in figure 2, $OE$ and $OD$ bisect $\angle AOC$ and $\angle BOC$ respectively. If $\angle BOC = 40^\circ$, find the measure of $\angle EOD$.
|
[
"\\boxed{45}"
] |
2e40f3c0-752e-4d49-b5fc-c90cd5521fb8
|
[
"images/2e40f3c0-752e-4d49-b5fc-c90cd5521fb8-0.png"
] | null | 1,907
|
As shown in the figure, $AB=AD$, $AC=AE$, $BC=DE$, with point $E$ on $BC$. If $\angle EAC=\angle BAD$ and $\angle EAC=42^\circ$, what is the degree measure of $\angle DEB$?
|
[
"\\boxed{42}"
] |
be181b9e-5987-4a59-8039-f53c525b2211
|
[
"images/be181b9e-5987-4a59-8039-f53c525b2211-0.png"
] | null | 1,908
|
As shown in the figure, it is known that $AB$ is the diameter of $\odot O$, $\angle ACD$ is the angle subtended by arc $AD$ on the circumference, $\angle ACD=30^\circ$. What is the measure of $\angle DAB$ in degrees?
|
[
"\\boxed{60}"
] |
570ea72b-898f-46d4-9ecd-6712369b3b71
|
[
"images/570ea72b-898f-46d4-9ecd-6712369b3b71-0.png"
] | null | 1,909
|
As shown in the figure, $AB$ is the diameter of $\odot O$, and $CD$ is a chord of $O$ with $CD\perp AB$ at point $E$. $\angle BCO = \angle D$; If $BE=9\text{cm}$ and $CD=6\text{cm}$, what is the radius of $\odot O$?
|
[
"\\boxed{5}"
] |
d040baf3-3473-4478-9d0d-12a04a2c0562
|
[
"images/d040baf3-3473-4478-9d0d-12a04a2c0562-0.png"
] | null | 1,910
|
As shown in the figure, two lines $l_{1}$ and $l_{2}$ pass through point A(2,0) and intersect the y-axis at points B and C, respectively, where point B is above the origin and point C is below the origin. It is known that $AB=\sqrt{13}$. If $BC=4$, find the equation of line $l_{2}$.
|
[
"\\boxed{y=\\frac{1}{2}x-1}"
] |
86d4ef1e-a56c-4b2c-8208-a13ecb74d7f6
|
[
"images/86d4ef1e-a56c-4b2c-8208-a13ecb74d7f6-0.png"
] | null | 1,911
|
As shown in the figure, the graph of the linear function $y=k_1x+b$ intersects the graph of the inverse proportion function $y=\frac{k_2}{x}$ at points $A(1, 2)$ and $B(-2, n)$. If point $P$ is on segment $AB$, and the area ratio $S_{\triangle AOP} : S_{\triangle BOP} = 1 : 4$, find the coordinates of point $P$.
|
[
"\\boxed{\\left( \\frac{2}{5}, \\frac{7}{5} \\right)}"
] |
b27facfb-253e-4c99-9c5c-7fbaf46cd251
|
[
"images/b27facfb-253e-4c99-9c5c-7fbaf46cd251-0.png"
] | null | 1,912
|
As shown in the figure, in the Cartesian coordinate system, point O is the origin, and the vertex A of the rhombus OABC has coordinates (3, 4). Connect OB and AC to intersect at point H. Draw line $BD \parallel AC$ passing through point B to intersect the x-axis at point D. Find the equation of line BD.
|
[
"\\boxed{y=-2x+20}"
] |
5c397b56-315c-4541-9bf5-d98bb07d0496
|
[
"images/5c397b56-315c-4541-9bf5-d98bb07d0496-0.png"
] | null | 1,913
|
As shown in the figure, it is known that the parabola $y=ax^2+2x+c$ intersects the x-axis at point $A(-1,0)$ and point $B$, and intersects the y-axis at point $C(0,3)$. Find the equation of the parabola, and calculate the coordinates of point $B$.
|
[
"\\boxed{B\\left(3, 0\\right)}"
] |
71551635-6a41-434c-8293-4966cd819cc1
|
[
"images/71551635-6a41-434c-8293-4966cd819cc1-0.png"
] | null | 1,914
|
Synthesis and Exploration: As shown in the figure, in the Cartesian coordinate system, the graph of the quadratic function $y=x^2+bx+c$ passes through point $A\left(0, -\frac{7}{4}\right)$ and point $B\left(1, \frac{1}{4}\right)$. Point $P$ is an arbitrary point on the graph of this function, with the x-coordinate $m$. A line segment $PQ$ is drawn parallel to the x-axis from point $P$, and the x-coordinate of point $Q$ is $-2m+1$. It is known that point $P$ and point $Q$ do not coincide, and the length of segment $PQ$ decreases as $m$ increases. What is the range of values for $m$?
|
[
"\\boxed{m<\\frac{1}{3}}"
] |
e731a965-03e9-40b8-9750-3c43bf199e18
|
[
"images/e731a965-03e9-40b8-9750-3c43bf199e18-0.png"
] | null | 1,915
|
As shown in the figure, in the square OABC, point O is the origin, point $C(-3, 0)$, point A is on the positive y-axis, points E and F are respectively on BC and CO, with $CE=CF=2$. The graph of the linear function $y=kx+b\ (k\ne 0)$ passes through points E and F, intersecting the y-axis at point G. The graph of the inverse proportion function $y=\frac{m}{x}\ (m\ne 0)$ through point E intersects AB at point D. Is there a point P on segment EF such that $S_{\triangle ADP}=S_{\triangle APG}$? If so, find the coordinates of point P; if not, please explain why.
|
[
"\\boxed{P\\left(-\\frac{4}{3}, \\frac{1}{3}\\right)}"
] |
08771d35-9bbe-4b29-984a-7e46039168d5
|
[
"images/08771d35-9bbe-4b29-984a-7e46039168d5-0.png"
] | null | 1,916
|
In square $ABCD$, $E$ and $F$ are points on sides $AD$ and $CD$ respectively, where $AE=ED$ and $DF=\frac{1}{4}DC$. Extend line $EF$ to intersect the extension of $BC$ at point $G$. $\triangle ABE\sim \triangle DEF$; if the side length of the square is $6$, what is the length of $BG$?
|
[
"\\boxed{15}"
] |
d6b96c0e-01e8-4657-bc21-5bd38104184c
|
[
"images/d6b96c0e-01e8-4657-bc21-5bd38104184c-0.png"
] | null | 1,917
|
In quadrilateral \(ABCD\), triangles \(OAB\) and \(OCD\) share the common vertex \(O\), and \( \triangle OAB \cong \triangle OCD\). As shown in Figure 2, points \(B\), \(O\), and \(C\) are not on the same line, and \(\angle AOB=90^\circ\), given that \(AD^2+BC^2=50\), \(AO=3\), find the area of \(\triangle OAB\).
|
[
"\\boxed{6}"
] |
a2af14ff-daa2-4f2a-a91f-28331e4008d3
|
[
"images/a2af14ff-daa2-4f2a-a91f-28331e4008d3-0.png"
] | null | 1,918
|
In the figure, within $\triangle ABC$, $\angle BAC=90^\circ$, $\angle ABC=60^\circ$, where $AD$ and $CE$ bisect $\angle BAC$ and $\angle ACB$, respectively. What is the degree measure of $\angle AOE$?
|
[
"\\boxed{60}"
] |
56a0a7e9-110e-426a-a5dd-cbc80cda5c23
|
[
"images/56a0a7e9-110e-426a-a5dd-cbc80cda5c23-0.png"
] | null | 1,919
|
As shown in the figure, in rectangle $ABCD$, we have $AB=10$ and $AD=6$. Let point E be on side $BC$, and fold $\triangle CDE$ along $DE$ such that point C falls onto point F on side $AB$. What is the length of $AF$?
|
[
"\\boxed{8}"
] |
42f883bc-3c18-450f-9494-231c5f0c681f
|
[
"images/42f883bc-3c18-450f-9494-231c5f0c681f-0.png"
] | null | 1,920
|
As shown in the figure, points A and B are on the graph of the linear function $y=kx+1$, where point A is in the first quadrant and point B is on the x-axis. Point D is on the positive half of the x-axis, and the coordinates of point C are $(1, -2)$. Quadrilateral OADC is a rhombus. What is the value of $k$?
|
[
"\\boxed{1}"
] |
3408d390-9358-473a-b683-fd3315093429
|
[
"images/3408d390-9358-473a-b683-fd3315093429-0.png"
] | null | 1,921
|
As shown in the figure, there is a rectangular sheet of paper ABCD with point E on the side AB. Fold $\triangle BCE$ along the straight line CE, so that point B falls on the diagonal AC, denoted as point F. If $AB=4$ and $BC=3$, what is the length of $AE$?
|
[
"\\boxed{\\frac{5}{2}}"
] |
5bbf71b8-a7cd-4360-afa3-17bdac347758
|
[
"images/5bbf71b8-a7cd-4360-afa3-17bdac347758-0.png"
] | null | 1,922
|
As shown in the figure, the graph of the quadratic function $y=ax^{2}+bx-3$ intersects the x-axis at points $A(-1,0)$ and $B(3,0)$, and intersects the y-axis at point $C$. The vertex of the parabola is point $M$. Point $P$ is a point on the parabola, with the x-coordinate of point $P$ being $m$ ($m>3$). Point $Q$ is on the axis of symmetry, and $AQ \perp PQ$. If $AQ=2PQ$, find the value of $m$.
|
[
"\\boxed{4}"
] |
04e59524-d512-4412-8f80-80a21d519090
|
[
"images/04e59524-d512-4412-8f80-80a21d519090-0.png"
] | null | 1,923
|
As shown in the figure, $AB=4$, $CD=6$, point $F$ is on $BD$, $BC$ and $AD$ intersect at point $E$, and $AB \parallel CD \parallel EF$. What is the length of $EF$?
|
[
"\\boxed{\\frac{12}{5}}"
] |
b8f8e9b8-014c-40ff-9906-b00424b5c0cc
|
[
"images/b8f8e9b8-014c-40ff-9906-b00424b5c0cc-0.png"
] | null | 1,924
|
As shown in the figure, it is known that line segment $AB=m$ (where $m$ is a constant), point $C$ is a point on the line $AB$ (not coinciding with $A$ or $B$), and points $P$, $Q$ are respectively on segments $BC$, $AC$, and satisfy $CQ=2AQ$, $CP=2BP$. As shown in Figure 1, with point $C$ on segment $AB$, what is the length of $PQ$ (expressed in terms of $m$)?
|
[
"\\boxed{\\frac{2m}{3}}"
] |
758abdee-6df8-4f63-a11e-7ab404d68d55
|
[
"images/758abdee-6df8-4f63-a11e-7ab404d68d55-0.png"
] | null | 1,925
|
As shown in the figure, in rectangle $ABCD$, $AB=9$, $AD=3$, and $M$, $N$ are points on $AB$, $CD$ respectively. When quadrilateral $MBCN$ is folded along $MN$, point $B$ coincides exactly with point $D$ and point $C$ with point $E$. Connect $BN$. Quadrilateral $DMBN$ is a rhombus; determine the length of the line segment $AM$.
|
[
"\\boxed{4}"
] |
b3b742c2-7843-43ff-9e00-a0538ada975b
|
[
"images/b3b742c2-7843-43ff-9e00-a0538ada975b-0.png"
] | null | 1,926
|
As illustrated in the rectangle $ABCD$, where $DC=6$, there exists a point $E$ on $DC$. When $\triangle ADE$ is folded along the line $AE$, such that point $D$ precisely falls on side $BC$, let this point be $F$. If the area of $\triangle ABF$ is 24, what is the length of $BF$?
|
[
"\\boxed{8}"
] |
c2e9f65d-064a-4122-b4b2-ea45f40496e0
|
[
"images/c2e9f65d-064a-4122-b4b2-ea45f40496e0-0.png"
] | null | 1,927
|
As shown in the figure, in rectangle $ABCD$, $DC=6$. There is a point $E$ on $DC$. By folding $\triangle ADE$ along the line $AE$, such that point $D$ falls exactly on the side $BC$, and let this point be $F$. If the area of $\triangle ABF$ is $24$, what is the length of $CE$?
|
[
"\\boxed{\\frac{8}{3}}"
] |
63ebf9c9-c890-4553-a475-45309b13bfd6
|
[
"images/63ebf9c9-c890-4553-a475-45309b13bfd6-0.png"
] | null | 1,928
|
In $\triangle ABC$, $AC=BC=10$, $\tan A = \frac{3}{4}$, points $D$ and $E$ are movable points on sides $AB$ and $AC$ respectively, and segment $DE$ is connected. Construct $\triangle ADE$ symmetric to $\triangle A'DE$ about $DE$. As shown in figure 1, when point $A'$ coincides with point $C$, what is the length of $DE$?
|
[
"\\boxed{\\frac{15}{4}}"
] |
f020fa41-5eda-4eeb-9878-4f41d38bb33c
|
[
"images/f020fa41-5eda-4eeb-9878-4f41d38bb33c-0.png"
] | null | 1,929
|
As shown in the figure, in right triangle $ABC$, the right-angle sides $AC=3\text{cm}$ and $BC=4\text{cm}$. Let $P$ and $Q$ be points moving along $AB$ and $BC$, respectively. Point $P$ moves from point $A$ towards point $B$ at a constant speed, while point $Q$ moves from point $B$ towards point $C$ at the same constant speed of $1\text{cm}$ per second. The movement of $P$ stops as soon as $Q$ reaches point $C$. Let the movement time of $P$ and $Q$ be $t$ seconds. For what value of $t$ is $\triangle PBQ$ an isosceles triangle with $\angle B$ as the vertex angle?
|
[
"\\boxed{\\frac{5}{2}}"
] |
e62afb27-3dc2-4d39-91e9-f5fcf295000a
|
[
"images/e62afb27-3dc2-4d39-91e9-f5fcf295000a-0.png"
] | null | 1,930
|
As shown in the figure, in $\triangle ABC$, $AB=AC=4$, $\angle B=\angle C=40^\circ$. Point D moves on line segment $BC$ (D does not coincide with B or C). Connect $AD$ and construct $\angle ADE=40^\circ$, with $DE$ intersecting line segment $AC$ at $E$. When $\angle BDA=100^\circ$, what is the value of $\angle DEC$?
|
[
"\\boxed{100}"
] |
bc9ba710-1d5b-4d46-849e-9b4a4f43555e
|
[
"images/bc9ba710-1d5b-4d46-849e-9b4a4f43555e-0.png"
] | null | 1,931
|
As shown in the figure, in $\triangle ABC$, $\angle ACB=90^\circ$, an arc is drawn with point B as the center and the length of BC as the radius, intersecting line segment AB at D; an arc is drawn with point A as the center and AD as the radius, intersecting line segment AC at point E, and CD is connected. If $BC=5$ and $AC=12$, what is the length of $AD$?
|
[
"\\boxed{8}"
] |
65e3d871-e3e4-483d-87d7-628926e2c25a
|
[
"images/65e3d871-e3e4-483d-87d7-628926e2c25a-0.png"
] | null | 1,932
|
As shown in the figure, in $\triangle ABC$, points D and E are located on sides AB and AC, respectively. CD and BE are connected, and it is given that BD = BC = BE. If $\angle A = 30^\circ$ and $\angle ACB = 70^\circ$, what are the measures of $\angle BDC$ and $\angle ACD$?
|
[
"\\boxed{20}"
] |
cbe62ab5-e724-41be-baa9-63c187515d93
|
[
"images/cbe62ab5-e724-41be-baa9-63c187515d93-0.png"
] | null | 1,933
|
As shown in Figure 1, in $\triangle ABC$, $BO\perp AC$ at point $O$, $AO=BO=3$, $OC=1$. A line is drawn through point $A$ such that $AH\perp BC$ at point $H$, and it intersects $BO$ at point $P$. Connect $OH$. What is the degree measure of $\angle AHO$?
|
[
"\\boxed{45}"
] |
0dafe787-3c10-46c8-90aa-a3462f61be88
|
[
"images/0dafe787-3c10-46c8-90aa-a3462f61be88-0.png"
] | null | 1,934
|
As shown in the figure, it is known that ship A is in the direction of 30 degrees north of east from lighthouse P, ship B is in the direction of 70 degrees south of east from lighthouse P, and ship C is on the bisector of $\angle APB$. What is the degree measure of $\angle APB$?
|
[
"\\boxed{80}"
] |
cf3bc834-268c-4424-bb1e-4c5a9339b1af
|
[
"images/cf3bc834-268c-4424-bb1e-4c5a9339b1af-0.png"
] | null | 1,935
|
As shown in the figure, it is known that point C is on the line segment AB, and M and N are the midpoints of AC and BC, respectively. Given the line segment DE, extend DE to F such that EF= $ \frac{1}{2}$DE; extend ED to G such that DG=2DE, with P and Q being the midpoints of EF and DG, respectively. If PQ=18 cm, what is the length of DE in cm?
|
[
"\\boxed{8}"
] |
0792c358-2f8c-47bd-9f2b-bba7558d1e28
|
[
"images/0792c358-2f8c-47bd-9f2b-bba7558d1e28-0.png"
] | null | 1,936
|
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