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<image>ABC conforms a sector. CBDE is geometrically a parallelogram. Arc FG is a half-circle. What is the arc length FG in the shape EDFG? | 77*π/2 |
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<image>ABCD is geometrically a square. DCE is an isosceles triangle, and the sides DC and CE are of equal length. Side FG is shaped into a semi-circle. What's the area of the entire shape ECFG? | 81*π/8 + 243/2 |
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<image>ABC is identified as a sector. CBDE takes the form of a parallelogram. EDFG conforms a square. Side HI stretches into an equilateral triangle within the rectangle. What is the total area of shape EGHJI? | 2345 - 1225*√(3)/4 |
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<image>(2009 Spring • Nanchuan District Final) As shown in the , in △ABC, AB=AC, D is a point on side AB, DE is perpendicular bisector of AC, ∠A=40°, then the degree measure of ∠BDC is ____. | 80 |
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<image>ABCD is identified as a parallelogram. DCE takes on a sector shape. ECF is geometrically a sector. Arc GH forms half of a circle. What is the perimeter length of FCGH? | 30*π + 162 |
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<image>The aluminum alloy window frame is shown in the figure (unit: cm). When 3a + 2b = 120 cm, the total length of the frame (excluding joints) is ___ cm.
| 456 |
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<image>ABCD conforms a square. CBEF conforms a square. What is the perimeter length of EBG? | 40 + 40*√(3) |
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<image>GFHI is a parallelogram. What is the measurement of IH in the shape GFHI? | 10*√(3)/3 |
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<image>ABC is identified as a sector. Side DE projecting into an equilateral triangle beyond the rectangle. FDGH is geometrically a square. What is the total boundary length of GDI? | 1 + √(3) |
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<image> The image is a park layout.
(1) Use coordinates to indicate the positions of the following attractions: Ethnic Culture Garden (____,____); __ Amusement Park (____,____); Zoo (____,____).
(2) The Zoo is in the ____ direction from the Ethnic Culture Garden; the Park Entrance is in the ____ direction from the Amusement Park.
(3) How do you get from the Park Entrance to the Aquarium? | \text{Ethnic Park |
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<image>As shown in the image, in the convex quadrilateral ABCD, the diagonals AC and BD intersect at point O. If the area of triangle AOD is 2, the area of triangle COD is 1, and the area of triangle COB is 4, then the area of quadrilateral ABCD is ____. | 15 |
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<image>Side FG being an equilateral triangle. HFI is geometrically a sector. In sector HFI, what is the length of arc HI? | 7*√(2)*π/2 |
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<image>As shown in the figure, in circle O, it is known that arc AB equals arc BC, and the ratio of arc AB to arc AmC is 3:4. Then the measure of ∠AOC is ___ degrees.
| 144 |
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<image>Side CD emerges as an equilateral triangle. FGH creates an equilateral triangle. HFI is an isosceles triangle, having HF and FI of the same length. Can you calculate the area of the entire shape IFJK? | 5782 |
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<image>CBD is an isosceles triangle where CB is the same length as BD. What is the total area measurement of the shape DBE? | 6*√(3) |
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<image>CDE makes up an equilateral triangle. DEFG is identified as a parallelogram. JKL shapes into an equilateral triangle. What is the measurement of the total area of IHJLK? | 6384 - 1764*√(3) |
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<image>Known: As shown in the figure, AB=20, C is the midpoint of AB, D is a point on CB, E is the midpoint of DB, and EB=3. (1) Find the length of CD; (2) Using C as the origin, please directly write down the numbers represented by A, B, D, E on the number line.
| 4 |
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<image>CDE is constructed as an equilateral triangle. DEF is geometrically a sector. FEGH forms a square. GEI is a sector. What is the total area measurement of the shape GEI? | 27*π/2 |
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<image>The figures A and B in the left image have the same perimeter. ____.(True/False) | \text{True |
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<image>As shown in the figure, in rectangle ABCD, the diagonals AC and BD intersect at point O, E is the midpoint of DC, EF is parallel to AC and intersects BD at F. If AD=3 and AB=4, then EF=______. | 1.25 |
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<image>As shown in the figure, in trapezoid ABCD, the area of triangle ABE is 60 square meters. The length of AC is 4 times that of AE. What is the area of trapezoid ABCD in square meters?
| 320 |
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<image>ABCD takes the form of a parallelogram. DCEF is a parallelogram. Can you calculate the area of the entire shape FEG? | 3025*√(3)/2 |
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<image>DCE is in the form of a sector. Side FG elongating into an equilateral triangle outside the rectangle. Side IJ leading into an equilateral triangle beyond the rectangle. How can we calculate the perimeter of GHIKJ? | 24 + 20*√(2) |
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<image>To celebrate the 90th anniversary of the founding of the Party and to beautify the community environment, a neighborhood is planning to build an art lawn. As shown in the figure, the lawn is composed of a series of congruent rhombuses that partially overlap each other, with all the longer diagonals in the same straight line. The intersection point of the diagonals of the previous rhombus is a vertex of the subsequent rhombus , such as rhombuses ABCD, EFGH, CIJK… Each rhombus has diagonals measuring 4m and 6m respectively.
(1) If the total area of the lawn is 39m², ____ such rhombuses are needed;
(2) If there are n such rhombuses (n ≥ 2, and n is an integer), the total area of this lawn is ____m². | 4 |
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<image>Side CD extending into an equilateral triangle beyond the rectangle. DEFG is geometrically a parallelogram. HIJ forms an equilateral triangle. Can you calculate the area of the entire shape GFHJI? | 7812 - 1764*√(3) |
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<image> As shown in the figure, in $$\triangle ABC$$, follow these steps to construct the figure:
$$①$$ With $$B$$ and $$C$$ as centers, and a radius greater than $$ \dfrac {1}{2}BC$$, draw arcs that intersect at points $$M$$ and $$N$$;
$$②$$ Draw the straight line $$MN$$ which intersects $$AB$$ at point $$D$$, and connect $$CD$$. If $$CD=AC$$ and $$∠B=25^{\circ}$$, then the measure of $$∠ACB$$ is ______. | 105 |
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<image>There are various positional relationships between two circles. The positional relationship that does not exist in the diagram is _______________.
| intersecting |
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<image>ABC is an isosceles triangle where AB = BC. FGH is constructed as an equilateral triangle. GHIJ forms a square. Can you tell me the length of IH in shape GHIJ? | 21 |
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<image>ABCD is a square. DCEF is a parallelogram. GHI is an equilateral triangle. What is the total length of the perimeter of FEGIH? | 110 |
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<image>As shown in the figure, in right triangle ABC, AB = CB and BO ⊥ AC. Triangle ABC is folded so that AB falls on AC, and point B coincides with point E on AC. After unfolding, the crease AD intersects BO at point F. Connect DE and EF. The following conclusions are given: ① There are 4 pairs of congruent triangles in the figure; ② If triangle DEF is folded along EF, point D does not necessarily fall on AC; ③ BD = BF; ④ The area of quadrilateral DFOE is equal to the area of triangle AOF. The correct number of the above conclusions is (___).
| 4 |
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<image> As shown in the figure, the diameter of circle A is 8, and the diameter of circle B is 6. Points A and B are both on the straight line m, and the diameter CD of circle A is perpendicular to the straight line m. When point B moves along the straight line m, let AB = d. If circle B moves to intersect with both circle A and CD, the range of values for d is ____. | 1<d<3 |
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<image>EFG is in the shape of an equilateral triangle. FGHI is identified as a parallelogram. IHJK forms a square. What is the perimeter of IHJK? | 264 |
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<image> The diameter of the Guomao revolving restaurant is 30m, and the width of the revolving part is 10m. What is the area of the revolving part in square meters? | 628 |
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<image>Side CD expanding into an equilateral triangle beyond the rectangle. What is the perimeter length of GFH? | 7*√(2) + 14 |
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<image>ABC is an isosceles triangle, having AB and BC of the same length. CBD is a sector. Square DBEF includes a circle inscribed within. How can we determine the area of the entire inscribed circle? | 400*π |
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<image>As shown in the figure, in the plane Cartesian coordinate system, given the points \(A(1,1)\), \(B(-1,1)\), \(C(-1,-2)\), and \(D(1,-2)\), point \(P\) starts from point \(A\) and moves counterclockwise along the sides of quadrilateral \(ABCD\) at a speed of 2 units per second; another point \(Q\) starts simultaneously from point \(C\) and moves clockwise along the sides of quadrilateral \(CBAD\) at a speed of 3 units per second. The coordinates of the 2017th meeting point are ________. | (0, -2) |
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<image>Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?\n | 2.5 |
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<image>ACD is an isosceles triangle, having AC and CD of the same length. Side EF is extended to a semi-circle inside the rectangle. What is the arc length EF in the shape DCEF? | 10*√(3)*π/3 |
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<image>As shown in the image, after a "typhoon," a flagpole was blown down from 9 meters above the ground. The tip of the fallen flagpole landed 12 meters away from the base of the pole. How tall was the flagpole before it was broken at least?
| 24 |
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<image>ABCD is geometrically a parallelogram. DCE is identified as a sector. Side FG is extended to form a semi-circle. What is the total area measurement of the shape ECFG? | 392*π + 1960*√(2) |
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<image>ABC is an isosceles triangle with AB having the same length as BC. CBD is in the form of a sector. What is the measurement of the total area of DBEF? | 754 |
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<image>ABCD is a square. DCEF is identified as a parallelogram. Side HI continuing into a semi-circle outside the rectangle. What is the length of HI in shape FGHI? | 20*√(3)*π |
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<image>As shown in the , it is known that △ABC is congruent to △DEB, with point E on AB, and DE intersects AC at point F,
(1) when DE = 8 and BC = 5, the length of segment AE is ______;
(2) Given ∠D = 35° and ∠C = 60°,
① find the measure of ∠DBC;
② find the measure of ∠AFD. | 3 |
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<image>CDE forms an equilateral triangle. HFIJ is identified as a parallelogram. How much is the perimeter of HFIJ? | 182*√(3)/3 + 158 |
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<image>ABC is an isosceles triangle in which AB equals BC in length. CBD is in the form of a sector. Side FG is extended to form a semi-circle. What is the overall area of the shape DEFG? | 16*π + 112*√(3) |
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<image>If the area of the yellow parallelogram is 36, the angle $\phi$ is vertical to $\beta$ and the angle $\alpha$ is vertical to $\gamma$, compute the degree of the angle marked with question mark. Round computations to 2 decimal places. | 33.03 |
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<image>Two set squares are overlapped as shown in the figure. If AB = 16 cm, then the area of the shaded part is ______ cm².
| 32 |
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<image>ABC is an isosceles triangle in which AB and BC are equal. CBD is an isosceles triangle where CB is the same length as BD. Side EF extends into a semi-circle. What is the total area measurement of the shape DBEF? | π/8 + 1/2 |
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<image>Side CD morphs into an equilateral triangle. FGH describes an equilateral triangle. JIKL is a parallelogram. In the shape JIKL, what is the measurement of LK? | 54 |
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<image>As shown in the diagram, in right triangle ABC, AB = AC, and D and E are points on the hypotenuse BC, with ∠DAE = 45°. After rotating △ADC clockwise by 90° about point A, △AFB is obtained. Connecting EF, the correct conclusions among the following are ___ . (fill in the number)
① ∠EAF = 45°;
② △ABE ∼ △CAD;
③ EA bisects ∠CEF;
④ BE^{2} + DC^{2} = DE^{2}.
| ①③④ |
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<image>ABC is in the form of a sector. CBD forms a sector. Side EF converting into a semi-circle outside. In shape DBEF, what is the length of EF? | π |
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<image>Side CD develops into an equilateral triangle. ECF is defined as a sector. GHI forms an equilateral triangle. Square IGJK contains a circle that is inscribed. How would you determine the perimeter of circle? | 55*π |
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<image>In triangle ABC, ∠ACB=90°, CD⊥AB at point D, let AC=b, BC=a, AB=c, CD=h.
Prove that: (1) \frac{}{} + \frac{}{} = \frac{}{};
(2) Can the sides with lengths a+b, h, and c+h form a triangle? If they can form a triangle, try to determine the shape of the triangle; if they cannot form a triangle, try to explain the reason.
| \frac{1 |
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<image>ABC is in the form of a sector. CBD is an isosceles triangle where the length of CB equals the length of BD. DBEF conforms a square. Side GH extends inward as a semi-circle inside the rectangle. What is the perimeter of DFGH? | 13*π/2 + 33 |
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<image>Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$? | 15 |
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<image>ABCD is a parallelogram. Side EF stretching into an equilateral triangle outside the rectangle. What would the perimeter of IHJ be? | 11 + 11*√(3) |
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<image> As shown in the figure, △ABC is an equilateral triangle with a side length of 2 cm. Point D is the midpoint of BC. △AEB is obtained by rotating △ADC around point A by 60°. Then ∠ABE = (___) degrees; BE = (___) cm. If DE is connected, then △ADE is a (___) triangle. | 60 |
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<image>CBD takes on a sector shape. DBE is an isosceles triangle in which DB and BE are equal. What is the base length DE in the isosceles triangle DBE? | -36*√(2) + 36*√(6) |
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<image>ABC is an isosceles triangle in which AB and BC are equal. CBDE conforms a square. EDFG is geometrically a square. HIJ shapes into an equilateral triangle. What is the full area of the shape EGHJI? | √(3)/4 + 1 |
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<image> As shown in the figure, AB∥CD, point E is on AB, and ∠1=∠2, ∠CED=58°. Find out the measure of ∠BCD when DE∥BC. Please explain the reasoning. | 119 |
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<image>As shown in the figure, in rectangle $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Points $$E$$ and $$F$$ are the midpoints of $$AO$$ and $$AD$$, respectively. If $$AB=6 \, \text{cm}$$ and $$BC=8 \, \text{cm}$$, then the perimeter of $$\triangle AEF$$ is ________.
| 9 |
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<image>There is a pond with a water surface width of $$10$$ feet. In the exact center of the pond, there is a reed that extends $$1$$ foot above the water surface. If this reed is pulled toward the midpoint of one side of the pond, its top just reaches the water surface at the edge of the pond. What are the depth of the water and the length of the reed in feet? | 12 |
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<image>ABC takes the form of a sector. CBD is in the form of a sector. DBEF conforms a parallelogram. What is the perimeter length of DBEF? | 42 |
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<image>As shown in the figure, AD bisects the exterior angle ∠EAC of △ABC, and AD is parallel to BC. If ∠BAC = 80°, then ∠B = ___ °. | 60 |
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<image> Given: As shown in the figure, point D is a point on AB. Triangle ADC and triangle BDC are isosceles triangles with bases AC and BC respectively. Find the degree measure of ∠ACB. | 90 |
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<image>As shown in the figure, if point C is the midpoint of line segment AB, and point D is on line segment CB, with DA = 6 and DB = 4, then the length of CD is ____. | 1 |
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<image>As shown in the figure, point C is on circle O. The central angle ∠AOB is rotated counterclockwise about point O to ∠A'OB'. The rotation angle is α (0° < α < 180°). If ∠AOB = 30° and ∠BCA' = 40°, then ∠α = _____°. | 110 |
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<image>Solve for the area of the shaded region in two different ways (unit: dm).
| 12 |
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<image>If the ABCD shape is a rectangle where a semi-circle has been removed from one side of it, the perimeter of the ABCD shape is 98, the BCFG shape is a rectangle where a semi-circle has been removed from one side of it, the length of the CF side is 10 and the perimeter of the BCFG shape is 64, compute the length of the AB side of the ABCD shape. Assume $\pi=3.14$. Round computations to 2 decimal places. | 24.39 |
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<image> As shown in the figure, in △ABC, ∠B=90°, AB=3, AC=5. Fold △ABC such that point C coincides with point A, and the fold line is DE. Then the perimeter of △ABE is (____ ). | 7 |
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<image>As shown in the figure, EF∥AD, ∠1 = ∠2, and ∠BAC = 70°. Find ∠AGD.
| 110 |
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<image>(2015 Spring • Yining City School Monthly Exam) As shown in the figure, in the rectangle ABCD, line segments AC and BD intersect at point O. DE is parallel to AC, and CE is parallel to BD. BC = 2 cm. Then, triangle EDC can be regarded as obtained by translating ____, and connecting OE, then OE = ____ cm. | OAB |
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<image>(2010 Autumn • Maanshan Final) As shown in the figure, the cross section of a river embankment is a trapezoid with BC ∥ AD, and the length of the slope AB facing the water is 13 meters with a slope ratio of 1:2.4. Then the height of the embankment BE is ____ meters. | 5 |
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<image>As shown in the figure, $$∠AOC$$ and $$∠DOB$$ are both right angles, and if $$∠DOC=35^{\circ}$$, then the measure of $$∠AOB$$ is __________. | 145^\circ |
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<image>As shown in the , the bisectors of ∠ABD and ∠BDC intersect at point E, and BE intersects CD at point F. ∠1 + ∠2 = 90°, and ∠2 = 40°. Find the measure of ∠3. | 50 |
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<image> As shown in the image, there are two roads from village A to village E (one passing through villages B, C, and D, and the other not passing through them). Which road is shorter? (The two roads are composed of one large semicircle and four identical small semicircles, respectively.) | Same length |
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<image>As shown in the diagram, AB is the diameter of circle O, and C and D are points on circle O. Given that ∠D = 35°, the measure of ∠BOC is ___ degrees.
| 110 |
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<image>A passenger ship encountered distress while sailing and sent out a distress signal. At location $$A$$, there was a rescue boat $$B$$ located 10 nautical miles to the south-west at an angle of $$45^{\circ}$$. Upon receiving the distress signal, the rescue boat immediately set out to search and found that the distressed passenger ship was heading south-east at an angle of $$75^{\circ}$$ and was approaching a small island at a speed of 9 nautical miles per hour. It is known that the maximum speed of the rescue boat is 21 nautical miles per hour.
$$(1)$$ To catch up with the passenger ship in the shortest possible time, determine the time required for the rescue boat to catch up with the passenger ship;
$$(2)$$ If the two boats meet at point $$C$$ in the shortest time, as shown in the figure, find the sine of angle $$B$$ in the triangle $$\triangle ABC$$. | \frac{2 |
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<image>DCEF is geometrically a parallelogram. FEGH is geometrically a square. In shape FEGH, how long is GE? | 88 |
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<image>As shown in the , it is known that point D is on line AB, point E is on line AC, lines BE and CD intersect at point F. If ∠A = 50°, ∠ACD = 40°, and ∠ABE = 28°, then the measure of ∠CFE is (____ ). | 62 |
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<image>As shown in the figure, in triangle ABC, AD is perpendicular to BC at D. By adding another condition ____, triangle ABD can be determined to be congruent to triangle ACD. | \text{BD=DC |
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<image>(Autumn 2014, Yancheng school-level final) As shown in the figure, in △ABC, ∠B=90°, AB=5, BC=12, AC=13. The distance from point A to the line on which BC lies is ____. | 5 |
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<image>ABCD takes the form of a parallelogram. EFG forms an equilateral triangle. FGHI conforms a square. HGJK forms a square. What is the entire perimeter length of HGJK? | 4 |
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<image> As shown in the diagram, in triangle △ABC, \( AB = AC \), and \( DE \) is the perpendicular bisector of \( AB \). \( D \) is the foot of the perpendicular, intersecting \( AC \) at \( E \). If \( AB = a \) and the perimeter of △ABC is \( b \), find the perimeter of △BCE. | b - a |
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<image>ABC takes on a sector shape. CBDE is identified as a parallelogram. EDF is an isosceles triangle, with ED equalling DF. Square FDGH includes an inscribed circle. In shape FDGH, how long is GD? | 6 |
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<image>Find the area of the shaded region (unit: decimeters).
| 30 |
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<image>DCE is an isosceles triangle where DC = CE. What's the perimeter of ECFG? | 6 |
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<image>
As shown in the figure, in rectangle \(ABCD\), the side \(BC=x\) and \(DC=y\). Connect \(BD\) and let \(CE \perp BD\), \(CE=2\), \(BD=4\). Then the value of \((x+y)^{2}-3xy+2\) is ______. | 10 |
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<image> As shown in the figure, AD and AE are respectively the altitude and angle bisector of △ABC. ∠C = 36°, ∠B = 72°, then the measure of ∠DAE is ____ (degrees). | 18 |
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<image>As shown in the figure, in $$\triangle ABC$$, $$∠A=36^{\circ}$$, $$AB=AC$$, $$BD$$ bisects $$∠ABC$$, and $$DE$$ is parallel to $$BC$$. How many isosceles triangles are there in the figure? | 5 |
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<image>$$(1)$$ As shown in the figure, in $$\triangle ABC$$, point $$D$$ is on $$BC$$, $$∠BAC=70^{\circ}$$, $$∠2=2∠3$$, $$∠1=∠C$$. Find the degree of $$∠2$$.
$$(2)$$ It is known that the sum of the interior angles of a polygon is $$4$$ times the sum of its exterior angles plus an additional $$180^{\circ}$$. Find the number of sides of this polygon. | 44^{\circ |
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<image>ABC is in the form of a sector. Side DE is designed as an equilateral triangle. FDG is an isosceles triangle where FD = DG. GDHI is a square. What is the measurement of HD in the shape GDHI? | 3 |
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<image> As shown in the figure, BC=3cm, AB=AC=4cm, ∠A=40°, point A and point B are symmetric with respect to line l, AC intersects line l at point D, then ∠C=(____), the perimeter of △BDC is (_____ ). | 70° |
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<image>ABCD is geometrically a square. DCE is an isosceles triangle with DC having the same length as CE. Side FG arching into a semi-circle outside. What is the arc length FG in the shape ECFG? | 11*π/2 |
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<image>Given: As shown in the figure, AD ∥ BC, and ∠1 = ∠2. Prove that ∠3 + ∠4 = 180°.
| 180° |
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<image>Problem: Given that line segments AB and CD intersect at point O, with AB = CD. Connect AD and BC. Please add a condition to ensure that △AOD ≌ △COB.
Xiaoming's Approach and Thought Process:
Xiaoming added the condition: ∠DAB = ∠BCD. His logic is: Drawing diagrams ① and ② with two situations, in both:
Draw straight lines DA and BC, which intersect at point E.
Given ∠DAB = ∠BCD, we derive that ∠EAB = ∠ECD.
Since AB = CD and ∠E = ∠E,
Thus, △EAB ≌ △ECD. Hence, EB = ED and EA = EC.
In diagram ①, ED - EA = EB - EC, i.e., AD = CB.
In diagram ②, EA - ED = EC - EB, i.e., AD = CB.
Also since ∠DAB = ∠BCD, and ∠AOD = ∠COB,
Thus, △AOD ≌ △COB.
Math Teacher's Viewpoint:
(1) The math teacher said that Xiaoming's added condition is incorrect. Please give an explanation.
Your Thoughts:
(2) Please add a new condition that satisfies the problem's requirements and provide a justification. | OA=OC |
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<image> As shown in the figure, DE ⊥ AB at E, ∠A = 25°, ∠D = 45°. Find the degree measure of ∠ACB. | 110° |
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<image>ABC is an isosceles triangle where AB is the same length as BC. Side DE develops into an equilateral triangle. Side GH turns into an equilateral triangle. IGJ is an isosceles triangle in which IG and GJ are equal. What's the area of the entire shape IGJ? | 676*√(3) |
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<image> As shown in the figure, AD is the median of △ABC, ∠ADC=60°, BC=4. After folding △ADC along line AD, point C falls on the position of C'. What is the length of BC'? Please explain the reason. | 2 |
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<image>If the length of the AC side is $5x - 35$, the length of the AB side is $5x - 39.79$, the degree of the BAC angle is 70 and the degree of the BCA angle is 40, compute the length of the AB side of the ABC triangle. Round computations to 2 decimal places and round the value of the variable "x" to the nearest natural number. | 10.21 |
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