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<image> As shown in the figure, A, B, and C are three points on the circle O. ∠ACB=40°. Then, ∠ABO is equal to ______ degrees. | 50 |
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<image>ABC is an isosceles triangle where the lengths of AB and BC are equal. How long is the perimeter of FEG? | 84*√(3) + 252 |
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<image> As shown in the figure, in △ABC, AD is the altitude, AE is the angle bisector, ∠B = 20°, and ∠C = 60°. Find the measure of ∠EAD. | 20° |
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<image> As shown in the figure, AD∥BC.
① AD is the bisector of ∠EAC, ∠B=64°, can you calculate the degree of ∠C?
② If ∠B=∠C, is AD the bisector of ∠EAC? Why? | 64° |
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<image>ABC is defined as a sector. EDFG conforms a square. How much is the perimeter of EDFG? | 260 |
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<image>ABCD is a square. CBEF is geometrically a square. CFGH conforms a parallelogram. Could you determine the area of the entire shape CFGH? | 169*√(6)/2 |
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<image>ABC is identified as a sector. CBD is an isosceles triangle where CB = BD. FEGH takes the form of a parallelogram. What is the side length HG in the shape FEGH? | 100 |
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<image>ABCD is a square. DCEF is geometrically a square. DFGH conforms a square. Side IJ converts into a semi-circle. Can you calculate the area of the entire shape DHIJ? | 162*π + 864 |
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<image>DCEF is a square. FEG is defined as a sector. GEH is an isosceles triangle where the sides GE and EH are identical in length. How long is the base GH in the isosceles triangle GEH? | 35*√(2 - √(2)) |
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<image>As shown in the figure, in $$\triangle ABC$$, $$∠C=90^{\circ}$$, $$∠B=30^{\circ}$$, the perpendicular bisector of $$AB$$, named $$ED$$, intersects $$AB$$ at point $$E$$ and $$BC$$ at point $$D$$. If $$CD=3$$, then the length of $$BD$$ is ______. | 6 |
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<image>The diagram shows a square $P Q R S$ with area $120 \mathrm{~cm}^{2}$. Point $T$ is the mid-point of $P Q$. The ratio $Q U: U R=2: 1$, the ratio $R V: V S=3: 1$ and the ratio $S W: W P=4: 1$. What is the area, in $\mathrm{cm}^{2}$, of quadrilateral $T U V W$? | 67 |
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<image>CBDE takes the form of a parallelogram. EDFG is geometrically a square. Can you tell me the length of FD in shape EDFG? | 63 |
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<image>EFG is identified as an equilateral triangle. GEHI forms a square. How can we determine the area of the entire shape GEHI? | 49 |
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<image>As shown in the figure, △ABC and △ECD are both isosceles right triangles, and point C is on line AD. The extension of AE intersects BD at point P. Please identify a pair of congruent triangles in the figure and provide the proof of their congruence.
| △ACE ≌ △BCD |
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<image>Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip? | 2.7 |
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<image>ABC is geometrically a sector. CBDE is geometrically a square. Side FG becoming an equilateral triangle outside the rectangle. Side IJ constructs an equilateral triangle. How would you calculate the total area of the shape HFIKJ? | 121*√(3)/4 + 209 |
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<image>Exploration: As shown in Figure ①, in △ABC, ∠ACB = 90°, CD ⊥ AB at point D. If ∠B = 30°, the measure of ∠ACD is ____ degrees;
Extension: As shown in Figure ②, ∠MCN = 90°, ray CP is inside ∠MCN, points A and B are on CM and CN respectively. Through points A and B, draw AD ⊥ CP and BE ⊥ CP, with foot points D and E respectively. If ∠CBE = 70°, find the measure of ∠CAD;
Application: As shown in Figure ③, points A and B are on the sides CM and CN of ∠MCN, ray CP is inside ∠MCN, points D and E are on ray CP. Connect AD and BE. If ∠ADP = ∠BEP = 60°, then ∠CAD + ∠CBE + ∠ACB = ____ degrees.
| 30 |
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<image>As shown in the figure, a set of Tangram pieces was made from a square with side length $$4cm$$, forming the pattern shown in the figure. The area of the shaded part in the figure is ______ $$cm^{2}$$.
| 9 |
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<image>Given: In △ABC, the perpendicular bisectors of AB and AC intersect BC at points M and N, respectively. AB = 4, AC = 7, BC = 10. Find the perimeter of △AMN. | 10 |
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<image>Given in the triangle △ABC as shown in the figure, ∠C=90°, BC=8cm, AC:AB=3:5. Point P starts from point B and moves along BC towards point C at a speed of 2 cm per second, and point Q starts from point C and moves along CA towards point A at a speed of 1 cm per second. If P and Q start from B and C respectively at the same time, after how many seconds does △CPQ become similar to △CBA? | 2.4 |
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<image> As shown in the figure, in parallelogram $$ABCD$$, $$AB=10$$, $$AD=8$$, and $$AC⊥BC$$. Find the area of the parallelogram $$ABCD$$. | 48 |
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<image> The school’s ecological planting base fenced off a trapezoidal vegetable plot in the vacant land next to the wall (as shown in the figure). A total of 35 meters of fence was used. If each vegetable occupies 0.15㎡ of land, how many vegetables can be planted in this plot? | 580 |
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<image>ACDE forms a square. Side FG being an equilateral triangle. How would you calculate the total area of the shape AEFHG? | 24*√(3) |
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<image>If the area of the brown parallelogram is 96 and the perimeter of the yellow parallelogram is 58, compute the degree of the angle marked with question mark. Round computations to 2 decimal places. | 64.16 |
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<image>ABC is defined as a sector. GHI forms an equilateral triangle. Can you calculate the perimeter of FEGIH? | 2*√(6) + 3*√(3) |
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<image>ABC is geometrically a sector. CBD forms a sector. DBEF is identified as a parallelogram. Side GH extends as an equilateral triangle inside the rectangle. How long is the perimeter of FEGIH? | 67 |
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<image>As shown in the figure, quadrilateral ABCD is an inscribed quadrilateral within a semicircle with center O and diameter AB. The diagonals AC and BD intersect at point E.
(1) Prove that △DEC ∼ △AEB;
(2) When ∠AED = 60°, find the area ratio of △DEC to △AEB. | △DEC ∽ △AEB |
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<image>ABC is defined as a sector. CBDE is a parallelogram. EDFG is a square. Could you determine the area of the entire shape EDFG? | 8100 |
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<image>ABC is an isosceles triangle with AB having the same length as BC. Side FG stretching into an equilateral triangle outside the rectangle. HFI is an isosceles triangle, having HF and FI of the same length. Could you provide the length of the base HI in the isosceles triangle HFI? | 90*√(3) |
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<image>ABCD conforms a parallelogram. DCEF is a square. FEGH forms a square. Can you calculate the area of the entire shape FEGH? | 1 |
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<image>If the diagonal of the pink rectangle is 24, compute the area of the blue triangle. Round computations to 2 decimal places. | 94.51 |
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<image>CDE exemplifies an equilateral triangle. FGH describes an equilateral triangle. Side IJ extending into an equilateral triangle beyond the rectangle. LMN makes up an equilateral triangle. How would you calculate the total area of the shape JKLNM? | 1764*√(3) + 5376 |
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<image>Granny Zhang has a parallelogram-shaped vegetable garden . If a fence is to be placed around the garden, please calculate the total length of the fence. | 192 |
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<image>ABC is an isosceles triangle, and the sides AB and BC are of equal length. CBD is geometrically a sector. DBE is identified as a sector. FGH shapes into an equilateral triangle. What is the total boundary length of EBFHG? | 170 |
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<image>Overlay two identical right-angled triangles as shown in the image. Find the area of the shaded region. (Units in the image are in centimeters.) | 40 |
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<image>ABCD conforms a square. CBE is an isosceles triangle where the length of CB equals the length of BE. GFH takes the form of a sector. In the sector GFH, what is the measurement of arc GH? | 6*π |
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<image>Rectangle $W X Y Z$ is cut into four smaller rectangles as shown. The lengths of the perimeters of three of the smaller rectangles are 11, 16 and 19 . The length of the perimeter of the fourth smaller rectangle lies between 11 and 19. What is the length of the perimeter of $W X Y Z$ ? | 30 |
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<image>ABC is a sector. Side FG extends inward as a semi-circle inside the rectangle. What is the full area of the shape EDFG? | 312 - 169*π/8 |
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<image>(2010 Spring•Nanhai District School Year-End Exam) As shown in the figure, D is a point on side BC of △ABC. If △ABD ∽ △CBA, then the additional condition required is ____. | ∠BAD = ∠C |
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<image>ABCD is identified as a parallelogram. DCEF is a parallelogram. Side GH creating an equilateral triangle. HIJK forms a square. What is the area size of the entire shape HIJK? | 3136 |
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<image>ABCD is identified as a parallelogram. DCEF is identified as a parallelogram. Could you determine the area of the entire shape HGIJ? | 7 |
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<image>ABCD conforms a parallelogram. EFG exemplifies an equilateral triangle. How do you measure the perimeter of FGHI? | 44 |
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<image>EFG is classified as an equilateral triangle. How do you calculate the area of the shape FHI? | 19208*√(3)/3 |
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<image>ABC is an isosceles triangle, with AB equalling BC. DBEF is geometrically a parallelogram. FEG is identified as a sector. Could you determine the perimeter of FEG? | 14*√(3)*(3 + π)/3 |
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<image>CDE defines an equilateral triangle. Side GH extends into an equilateral triangle outside the rectangle. Side JK continuing into an equilateral triangle outside the rectangle. How would you calculate the total area of the shape HIJLK? | 5*√(3) |
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<image>ABC is a sector. CBDE is geometrically a parallelogram. EDFG is a square. Side HI forming a semi-circle beyond the rectangle. Could you determine the area of the entire shape GFHI? | 1225*π/2 + 3710 |
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<image>DCE takes the form of a sector. What is the length of arc DE in sector DCE? | 22*√(6)*π |
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<image>DCEF is a square. Side GH growing into an equilateral triangle outside the rectangle. How would you calculate the total area of the shape ECGIH? | 169*√(3)/4 + 182 |
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<image>The area of the shaded part shown in the figure below is ____ square centimeters.
| 54 |
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<image>Side CD forming an equilateral triangle. DEFG is geometrically a square. FEHI is geometrically a square. How much area does the shape FEHI cover? | 1 |
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<image>Side CD becomes an equilateral triangle. DEFG is a parallelogram. GFHI forms a square. What would the area of the entire shape GFHI be? | 1 |
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<image>ABC takes on a sector shape. CBDE conforms a square. CEF is an isosceles triangle where the lengths of CE and EF are equal. GHI makes up an equilateral triangle. What is the total perimeter measurement of FEGIH? | 37 |
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<image> As shown in the figure, it is known that the side length of the square \(ABCD\) is \(6\), \(E\) and \(F\) are points on sides \(AB\) and \(BC\), respectively, and \(\angle EDF = 45^{\circ}\). Rotate \(\triangle DAE\) counterclockwise by \(90^{\circ}\) about point \(D\) to obtain \(\triangle DCM.\) If \(AE = 2\), then the length of \(FM\) is ______. | 5 |
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<image>As shown in the figure, the area of the parallelogram is 32 square meters. The area of the shaded part is ____ square meters.
| 12.56 |
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<image>ACDE is identified as a parallelogram. EDF is an isosceles triangle where ED and DF are of equal length. Square FDGH contains a circle that is inscribed. What is the overall area measurement of the inscribed circle? | 81*π |
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<image>ABCD is geometrically a parallelogram. FEG takes the form of a sector. How do you find the area of the entire shape FEG? | 2401*π/3 |
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<image> As shown in the figure, triangle $$\triangle ABC$$ is translated in the direction of $$BC$$ to obtain triangle $$\triangle DEF$$. If $$∠B=90^{\circ}$$, $$AB=6$$, $$BC=8$$, $$BE=2$$, $$DH=1.5$$, the area of the shaded region is ______. | 10.5 |
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<image>DCEF is a parallelogram. FEGH is a square. Could you determine the area of the entire shape FEGH? | 16 |
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<image>ABC is an isosceles triangle where AB is the same length as BC. CBDE forms a square. CEFG is a square. How would you determine the perimeter of CEFG? | 52 |
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<image>Side DE becoming an equilateral triangle outside the rectangle. EFGH is geometrically a square. In the shape EHI, what is the measurement of EI? | 8*√(6) |
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<image> As shown in the figure, AB∥CD, BE intersects CD at point F, ∠B=45°, ∠E=21°. Then ∠D is ____. | 24 |
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<image>Given that ∠1 = ∠2, please add a condition ______ to make △ABC ∼ △ADE. | ∠B=∠D |
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<image>ABCD is identified as a parallelogram. DCEF conforms a parallelogram. FEG conforms a sector. How long is arc FG in sector FEG? | 48*π |
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<image>As shown in the figure, ∠BAC=30°, AB=10. Now please provide the length of segment BC, such that triangle △ABC can be uniquely determined. You think the length of BC can be ___.
| 5 |
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<image>Each side of a triangle $A B C$ is being extended to the points $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}$ and $\mathrm{U}$, so that $\mathrm{PA}=\mathrm{AB}=\mathrm{BS}, \mathrm{TC}=\mathrm{CA}$ $=\mathrm{AQ}$ and $\mathrm{UC}=\mathrm{CB}=\mathrm{BR}$. The area of $\mathrm{ABC}$ is 1. How big is the area of the hexagon PQRSTU? | 13 |
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<image>As shown in the , it is known that $$AB/\!/CD$$, then the relationship between $$∠1$$ and $$∠2$$, $$∠3$$ is ______ . | ∠1=∠2+∠3 |
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<image> As shown in the figure, line AB is parallel to line CD, and GH intersects AB and CD at points M and F respectively. If ∠GMB = 70° and ∠CEF = 50°, then ∠C = ____. | 20 |
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<image> As shown in the figure, in the right trapezoid ABCD, triangles ABE and CDE are both isosceles right triangles, and BC = 20 cm. Then the area of the right trapezoid ABCD is ____ cm^{2}. | 200 |
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<image>Given that the perimeter of △ABC is 50 cm, the midline DE = 8 cm, the midline DF = 10 cm, then the length of the other midline EF is ______ cm. | 7 |
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<image>As shown in the diagram, in △ABC, D is a point on BC, ∠C = 62°, ∠CAD = 32°. What is the measure of ∠ADB?
| 94 |
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<image>As shown in the figure, in the equilateral triangle △ABC, point D and point E are located on sides AB and AC, respectively, and DE is parallel to BC. If BC = 8 cm and AD:DB = 1:3, then the perimeter of △ADE is _______ cm. | 6 |
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<image>DCE is an isosceles triangle where DC = CE. ECF is an isosceles triangle where EC and CF are of equal length. How much is the area of the entire shape ECF? | √(3)/4 |
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<image>DCEF is a square. DFGH is geometrically a square. GFI is a sector. What is the side length IF in the shape GFI? | 12 |
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<image>In the diagram, in ∠AOB, OD is the bisector of ∠BOC, and OE is the bisector of ∠AOC. Given that ∠AOB=140, then ∠EOD=___________ degrees. _ | 70 |
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<image>FEG is an isosceles triangle where FE and EG are of equal length. GEH is in the form of a sector. What is the arc length GH in the sector GEH? | 64*π |
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<image>Divide a circle with a diameter of 4 centimeters into several equal parts, then cut it and reassemble it as shown in the image. The circumference of the reassembled shape is ____ centimeters more than the original circle's circumference.
| 4 |
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<image>Given: Point D is on the extension of side BC of triangle ABC, DF is perpendicular to AB at point F, intersects AC at point E, ∠A = 30°, ∠D = 20°, find the measure of ∠ACB.
| 80 |
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<image>As shown in the figure, △ABC is rotated around point A in a counterclockwise direction by 60° to obtain △AB′C′. Then △ABB′ is a ______ triangle.
| equilateral |
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<image>As shown in the figure, given that ∠1 = ∠2 and ∠3 = 73°, find the measure of ∠4 in degrees.
| 107 |
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<image>ABC takes on a sector shape. CBDE is identified as a parallelogram. EDF is an isosceles triangle in which ED and DF are equal. In the shape EDF, what is the measurement of FD? | 11 |
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<image>As shown in the figure, the diagonals AC and BD of rectangle ABCD intersect at point O. CE is perpendicular to BD, and DE:EB = 3:1. OF is perpendicular to AB at F, with OF = 3. Find the length of the diagonals of the rectangle. | 12 |
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<image>ABC is an isosceles triangle where the length of AB equals the length of BC. CBD is an isosceles triangle where CB and BD are of equal length. DBE is in the form of a sector. EBFG is geometrically a square. What is the measurement of the total area of EBFG? | 9 |
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<image>ABC forms a sector. CDE is an isosceles triangle where CD and DE are of equal length. How long is ED in shape CDE? | 8*√(2) |
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<image>ABCD takes the form of a parallelogram. DCE is defined as a sector. Side FG growing into an equilateral triangle outside the rectangle. Square GHIJ has an inner inscribed circle. Could you specify the length of JI in shape GHIJ? | 2 |
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<image>DCEF is geometrically a square. ECG is an isosceles triangle, having EC and CG of the same length. GCH is an isosceles triangle where GC is the same length as CH. In isosceles triangle GCH, what is the measure of angle CGH and angle GHC? | 45 |
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<image> The image shows a rectangular lawn with a 5m wide path in the middle. Find the area of the path. (Unit: m) | 425 |
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<image>The side length of square A in the image is 12 centimeters, and the side length of square B is 8 centimeters. What is the perimeter of the large square in centimeters?
| 80 |
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<image>DCEF is geometrically a square. Square ECGH includes a circle inscribed within. What is the total surface area of the circle? | 169*π |
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<image>ABC is an isosceles triangle where the length of AB equals the length of BC. CBD conforms a sector. Square DBEF has an inscribed circle. What is the overall area of the inscribed circle? | 64*π |
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<image>ABCD takes the form of a parallelogram. DCEF conforms a square. Can you tell me the perimeter of HGIJ? | 14 |
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<image> Fold a rectangular piece of paper $$ABCD$$ along $$EF$$ such that the intersection of $$ED$$ and $$BC$$ is at point $$G$$. Points $$D$$ and $$C$$ are located at points $$M$$ and $$N$$ respectively after folding. Given that $$∠EFG=55^{\circ}$$, find:
$$(1) \text{The measure of} \, ∠FED;$$
$$(2) \text{The measure of} \, ∠FEG;$$
$$(3) \text{The measures of} \, ∠1 \, \text{and} \, ∠2.$$ | 55° |
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<image>If the gray shape is a combination of a rectangle and a semi-circle, the pink shape is a rectangle where an equilateral triangle has been removed from one side of it and the perimeter of the pink shape is 78, compute the perimeter of the gray shape. Assume $\pi=3.14$. Round computations to 2 decimal places. | 81.41 |
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<image>(2014 Spring • Jinhua school-level monthly exam) As shown in the figure, one side of △OAB, OB, lies on the line l. △OAB is translated 2 cm to the right along the line l to obtain △CDE. If CB = 1 cm, then OE = ____ cm. | 5 |
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<image>As shown in the figure, given that CB∥DE and AB∥EF, the number of angles in the figure that are equal to ∠B is ______. | 3 |
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<image>As shown in the figure, O is a point on line AB, OC bisects ∠BOD, OE is perpendicular to OC with O as the foot, what is the relationship between ∠AOE and ∠DOE? Please provide a reason. __ | ∠AOE=∠DOE |
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<image>Side CD extends beyond the rectangle into an equilateral triangle. Side FG becoming an equilateral triangle outside the rectangle. Square HFIJ has an inner inscribed circle. How do you find the perimeter of circle? | 3*π |
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<image>As shown in the figure, in triangle △ABC, DE is parallel to AB, and the ratio of CD to DA is 2:3. If DE is 4, then the length of AB is __. | 10 |
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<image>ACD is a sector. DCE forms a sector. ECF is a sector. How long is arc EF in sector ECF? | 8*π |
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<image>Calculate the area of the shaded part.
| 18 |
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<image>As shown in the figure, points A and B are 25 km apart on the railway, and C and D are two villages. DA is perpendicular to AB at A, and CB is perpendicular to AB at B. It is known that DA = 15 km and CB = 10 km. Now, a specialty product collection station E is to be built on the railway AB, so that the distances from villages C and D to station E are equal. How many kilometers away from station A should station E be built?
_ | 10 |
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