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<image>As shown in the figure , line AB is perpendicular to CD at point O, and line EF passes through point O, with ∠AOE equal to 40°. Then, ∠BOF is ____ degrees, and ∠DOF is ____ degrees. | 40 |
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<image>(2013 Autumn • School-level mid-term in Wuming County) Compare the perimeters of shapes A and B in the figure. The perimeter of shape A is ____ the perimeter of shape B. (Fill in with “>”, “<” or “=”) | = |
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<image>As shown in the figure, △ABC ≌ △DEF. Based on the information provided in the figure, write x = ___.
| 20 |
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<image> As shown in the figure, the closer the flagpole of the same height is to the streetlight, the more (__ ) its shadow becomes. | short |
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<image>Side GH being an equilateral triangle. In the shape FEGIH, what is the measurement of IG? | 13 |
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<image>ABC is an isosceles triangle where AB is the same length as BC. CBDE is a parallelogram. EDF is in the form of a sector. What is the overall perimeter of EDF? | 20*π + 80 |
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<image>As shown in the figure, \(AD\) is a diagonal of the regular pentagon \(ABCDE\). Then \(\angle BAD =\ \_\_\_\_\_ ^{\circ}\). | 72 |
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<image>As shown in the figure, in the quadrilateral ABCD, AD ∥ BC, E is the midpoint of CD, connect AE and BE, extend AE to intersect the extension of BC at point F.
(1) Prove: △DAE ≌ △CFE;
(2) If BE ⊥ AF, prove: AB = BC + AD. | △DAE≌△CFE |
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<image>As shown in the figure: point P lies within ∠AOB, and the points P1 and P2 are the reflections of point P about OA and OB, respectively. Connecting P1 and P2 intersects OA at M and OB at N. Given that P1P2 = 15, the perimeter of △PMN is ______; __ | 15 |
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<image>ABCD conforms a parallelogram. DCEF is geometrically a parallelogram. FEGH conforms a parallelogram. In the shape FEGH, what is the measurement of HG? | 3 |
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<image>If the arc length of the yellow sector is 15.42 and the angle $\alpha$ is vertical to $\beta$, compute the length of the side of the yellow sector marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places. | 9.82 |
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<image>ABCD forms a square. EBFG is geometrically a parallelogram. What is the side length GF in the shape EBFG? | √(3) |
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<image> As shown in the figure, a plane needs to fly from point A to point B. Due to strong winds, it deviates from the original route (AB) by 18° at the beginning (i.e., ∠A = 18°) and arrives at point C. It is known that ∠ABC = 10°. What angle should the plane now fly at to reach point B? (i.e., find the degree measure of ∠BCD) | 28° |
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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>As shown in the figure, lines AB and CD are cut by CE. When:
(1) ∠C = ____ , then AB ∥ CD;
(2) ∠BFC + ____ = 180°, then AB ∥ CD. | ∠EFB |
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<image>As shown in the figure, in triangle \( \Delta ABC \), \( \angle ACB = 90^\circ \), and CD is an altitude of \( \Delta ABC \). If \( \angle B = 28^\circ \), find the measure of \( \angle ACD \).
| 28 |
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<image>As shown in the figure, it is known that $$∠A=20^{\circ}$$, $$∠CBE=100^{\circ}$$, and $$∠C=40^{\circ}$$. Find the measure of $$∠ADE$$. | 20 |
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<image>As shown in the figure, in triangle △ABC, ∠C=90°, DB is the angle bisector of ∠ABC, point E is the midpoint of AB, and DE⊥AB. If BC=5 cm, then AB=__▲__cm. | 10 |
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<image> As shown in the figure, in the parallelogram ABCD, EF ∥ AC, and it intersects the extended lines of DA and DC at points E and F, and intersects AB and BC at points H and G, respectively. How many triangles similar to △AEH (excluding congruence) are there in the figure? (___). | 4 |
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<image>Side CD converts into an equilateral triangle. DEFG conforms a square. FEH is an isosceles triangle in which FE equals EH in length. What is the side length JI in the shape HEIJ? | 13 |
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<image>DCE is identified as a sector. Arc FG takes the shape of a semi-circle. Could you specify the length of FG in shape ECFG? | π/2 |
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<image>ABCD is identified as a parallelogram. HGI is geometrically a sector. How do you measure the perimeter of HGI? | 22*π + 132 |
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<image>As shown in the figure, in the Cartesian coordinate plane, the right triangle $$Rt\triangle ABC$$ is located in the first quadrant. The legs $$BC$$ and $$BA$$ are parallel to the $$y$$ axis and $$x$$ axis, respectively. The coordinates of point $$A$$ are $$(5,1)$$, $$BC=2$$, and $$AB=4$$.
$$(1)$$ If the graph of the inverse proportion function $$y=\dfrac{a}{x}$$ ($$x > 0$$) passes through point $$C$$, then $$a=$$________. $$(2)$$ Find the equation of the line on which $$AC$$ lies. $$(3)$$ How many units does $$\triangle ABC$$ need to be translated to the right for point $$B$$ to fall on the hyperbola in $$(1)$$? $$(4)$$ If the graph of the inverse proportion function $$y=\dfrac{a}{x}$$ ($$x > 0$$) intersects any side of the $$\triangle ABC$$, directly write the range of values for $$a$$ ________________________________. | 3 |
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<image> As shown in the figure, in $$\triangle ABC$$, $$∠B=30^{\circ}$$, $$ED$$ is the perpendicular bisector of $$BC$$, and $$ED=3$$. Find the length of $$CE$$: ______ . | 6 |
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<image>DCE is an isosceles triangle where the sides DC and CE are identical in length. ECF is a sector. FCGH conforms a square. How much is the perimeter of FCGH? | 96 |
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<image>Calculate the area of the following shape. (Unit: cm)
| 24 |
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<image> Place a right triangle ABC and ADE as shown in the figure (where ∠B=60°, ∠E=45°). It is known that DE intersects AC at point F, and AE ∥ BC. Then, the measure of ∠AFD is ____. | 75 |
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<image>(Spring 2011, Nanjing School-level Final Exam) As shown in the figure, ∠ABD = ∠CAB. Please add a condition to make △DAB ≅ △CBA. The condition you add is ____ (just provide one condition). | ∠DAB=∠CBA |
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<image>ABCD takes the form of a parallelogram. DCEF is geometrically a parallelogram. How would you calculate the perimeter of FEGH? | 16 + 12*√(3) |
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<image>What fraction of the square is shaded?
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{5}{12}$
D. $\frac{3}{7}$
E. $\frac{1}{2}$ | E. $\frac{1}{2}$ |
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<image> "As shown in the figure, △A′B′C′ is obtained by translating △ABC. In this translation, the distance of translation is the length of segment AA′." Is this statement correct? ____. | No |
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<image>As shown in the image, points A and B have coordinates (1,4) and (4,4) respectively. The vertex of the parabola \(y = a(x - m)^2 + n\) moves along the line segment AB and intersects the x-axis at points C and D (with C to the left of D). If the minimum x-coordinate of point C is -3, then the maximum x-coordinate of point D is ___.
| 8 |
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<image>Compute the perimeter of the ABCD trapezoid. Round computations to 2 decimal places. | 79 |
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<image>(2016 Autumn • Xicheng District School-level Midterm) As shown in the figure, in right triangle ABC, ∠C=90°, the perpendicular bisector of AB intersects side BC at point E. Connect AE. If ∠B=15°, then ∠EAC=____. | 60° |
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<image>As shown in the figure (6), BD is the bisector of ∠ABC, DE⊥AB at E, \( S_{△ABC} = 45 \, \text{cm}^2 \), \( AB = 18 \, \text{cm} \), \( BC = 12 \, \text{cm} \). Then \( DE = \) ____ cm. | 3 |
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<image>In the triangle $W X Y$ points $Z$ on $X Y$ and $T$ on $W Z$ are, as shown on the right. If one connects $\mathrm{T}$ with $\mathrm{X}$, a figure with nine internal angles is created as shown in the figure on the right. From those 9 angles, what is the smallest number that could be a different size to each other | 3 |
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<image>As shown in the figure, among the following rectangles I, II, and III, the similar rectangles are (___). | I and III |
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<image>ABC is an isosceles triangle in which AB equals BC in length. What is the length of EB in shape DBE? | 12 |
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<image>As shown in the figure, AB = AC. The perpendicular bisector of AB intersects AB at point E and intersects AC at point D. Given that ∠A = 40°, find ∠DBC = ______°. | 30 |
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<image>(Spring 2012•Jinan Final Exam) As shown in the figure, in △ABC, AB = AC, ∠A = 50°, and P is a point within △ABC such that ∠PBC = ∠PCA. Then ∠BPC = ____. | 115 |
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<image>As shown in the figure, in the square ABCD, AB = 1, and point P is a point on the diagonal AC. Squares are constructed with AP and PC as their respective diagonals. The sum of the perimeters of the two smaller squares is _______.
| 4cm |
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<image>ABC forms a sector. Can you tell me the length of FE in shape CDEF? | 2 |
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<image>Side CD extends beyond the rectangle into an equilateral triangle. ECFG forms a square. FCHI is identified as a parallelogram. In shape FCHI, what is the length of IH? | 9 |
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<image>ABCD is geometrically a parallelogram. DCE is an isosceles triangle where the length of DC equals the length of CE. ECFG takes the form of a parallelogram. Side HI is extended to a equilateral triangle inside the rectangle. How much area does the shape GFHJI cover? | 5400 - 1296*√(3) |
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<image>As shown in the figure, C and D are two points on the line segment AB. If CB = 4 cm, DB = 7 cm, and D is the midpoint of AC, find the length of AB. | 10 |
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<image>DCE takes the form of a sector. What is the entire perimeter length of ECFG? | 60 + 48*√(3) |
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<image>As shown in the figure, given that △ABC is inscribed in circle O, point D is on the extension of OC, AD is the tangent to circle O, and if ∠B = 30° and AC = 2, then the length of OC is ___.
| 2 |
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<image>As shown in the figure, △DEF is the corresponding figure of △ABC after being translated horizontally to the right. If ∠B = 31° and ∠C = 79°, then the measure of ∠D is ___ degrees. | 70 |
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<image>Square FEGH contains a circle that is inscribed. What is the total area of the circle? | 169*π/4 |
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<image>As shown in the figure, AD is the altitude from the base BC of the isosceles triangle ABC. DE is parallel to AB and intersects AC at point E. Determine whether △ADE is an isosceles triangle and explain the reason. | Yes |
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<image>(2015 Autumn • Longhai City Final Exam) As shown in the figure, lines a and b are intersected by line c. To ensure that a ∥ b, one necessary condition is ____. (Fill in one condition that you think is appropriate) | ∠3 = ∠1 |
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<image>As shown in the , it is known that $$AD$$ is the perpendicular bisector of $$BC$$ with the foot of the perpendicular being $$D$$, the perimeter of $$\triangle ABC$$ is $$32$$, the perimeter of $$\triangle ACD$$ is $$24$$. Then the length of $$AD$$ is ______ . | 8 |
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<image>Given, as shown in the figure, in △ABC, BC=8, the perpendicular bisector of AB intersects BC at D, the perpendicular bisector of AC intersects BC at E. What is the perimeter of △ADE?
| 8 |
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<image>ABCD conforms a square. DCEF forms a square. FEGH is a parallelogram. HGIJ conforms a square. Could you determine the area of the entire shape HGIJ? | 100 |
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<image>ABCD is a parallelogram. FEG is an isosceles triangle where FE and EG are of equal length. GEH is an isosceles triangle where GE is the same length as EH. Please provide the length of HE in shape GEH. | 2 |
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<image>Side CD forming an outward equilateral triangle beyond the rectangle. GFH is an isosceles triangle, and the sides GF and FH are of equal length. HFI is an isosceles triangle with HF having the same length as FI. How long is IF in shape HFI? | 3 |
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<image>As shown in the figure, in △ABC, ∠ACB = 90°, ∠A = 40°. A circle with center C and radius CB intersects AB at point D. Then ∠ACD = ___ degrees.
| 10 |
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<image>As shown in the figure, $$AB$$ is the diameter of circle $$⊙O$$, and $$C$$ and $$D$$ are two points on $$⊙O$$. If $$∠BCD=28^{\circ}$$, then $$∠ABD=$$ ______ $${\,\!}^{\circ}.$$ | 62 |
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<image>DCEF forms a square. Can you tell me the length of EG in shape ECG? | √(2) |
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<image>ABCD is identified as a parallelogram. DCEF conforms a square. DFG conforms a sector. What is the total perimeter measurement of DFG? | 2*π + 16 |
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<image> As shown in the figure, given that $$AB \parallel CD$$, $$AB = AC$$, and $$∠ABC = 68^{\circ}$$, find $$∠ACD =$$ ______ . | 44° |
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<image>The perimeter of an equilateral triangle is 1. Following a certain pattern, it transforms into the following figures. The perimeter of the fourth figure is ____.
| 1 |
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<image>【Problem】As shown in the image, in △ABC, the bisectors of ∠ABC and ∠ACB intersect at point O. A line DE is drawn through point O parallel to BC. If BD + EC = 5, then DE is equal to (___)
A. 7_______ B. 6_______ C. 5________ D. 4 | 5 |
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<image>ABC forms a sector. CBDE takes the form of a parallelogram. EDFG forms a square. EGH takes the form of a sector. What is the measurement of HG in the shape EGH? | 18 |
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<image>(2014 Autumn • Mid-term exam in Anlu City) As shown in the figure, the side length of the square ABCD is 4. A sufficiently large right-angled set square is placed with its right angle vertex at point A. The two perpendicular sides of the set square intersect CD at point F, and the extension line of CB at point E. The area of quadrilateral AECF is _____. | 16 |
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<image> As shown in the figure, in parallelogram ABCD, point E is on side BC, and BE:EC = 1:2. Connect AE and intersect BD at point F. The ratio of the area of △BFE to the area of △DFA is ___. | 1:9 |
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<image>As shown in the figure, AB ∥ CD, CB ∥ DE. If ∠B = 72°, then what is the measure of ∠D? (___)
| 108 |
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<image>As shown in the figure, \textit{DH}∥\textit{EG}∥\textit{BC} and \textit{DC}∥\textit{EF}, there are ______ angles equal to ∠1. | 5 |
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<image>If an exterior angle of an isosceles triangle is , then its base angle is ____ ________ | 35° |
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<image>(2016 Spring • Wuhou District Final Exam) As shown in the figure, in the right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), the perpendicular bisector of \( BC \) intersects \( AB \) at \( E \) with foot \( D \). Connecting \( CE \), if \( CE \) bisects \( \angle ACB \), then the measure of \( \angle B \) is ____. | 30 |
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<image>ABC is an isosceles triangle where AB and BC are of equal length. CBDE takes the form of a parallelogram. Can you calculate the perimeter of EDF? | 2*√(2) + 4 |
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<image>As shown in the figure, the coordinates of points A and B are (2,0) and (0,1) respectively. If the line segment AB is translated to \(A_{1}B_{1}\), and the coordinates of \(A_{1}\) and \(B_{1}\) are (3,1) and (a,b) respectively, then the value of \(a + b\) is _
| 3 |
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<image>Little Tour Guide_
(1) The little mouse walks ____ meters to the ____ and then walks ____ meters to the ____, arriving at the Science City.
(2) The little fox walks ____ meters to the ____ and then walks ____ meters to the ____, arriving at the Jingshan Park.
(3) The fox's home is to the ____ of the bookstore, and the Science City is to the ____ of Jingshan Park.
(4) ____'s home is closer to the bookstore. | Mouse's home |
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<image> Logarithm$$($$born around 250 AD$$)$$ is a great mathematician in the history of Chinese numerals and holds an important place in the history of world mathematics as well. His masterpieces $$《$$The Nine Chapters on the Mathematical Art$$》$$ and $$《$$The Sea Island Mathematical Manual$$》$$ are valuable mathematical heritages of China.
$$(1)$$ One of these books focuses entirely on the measurement of heights and distances, using tools connected by vertical relationships, such as measuring rods and horizontal bars. All problems utilize data obtained from two or more measurements to calculate the height, depth, width, and distance of otherwise unreachable targets. This book was included in the Yongle Encyclopedia, compiled during the reign of Emperor Yongle, and is currently preserved in the library of the University of Cambridge, UK. The title of this book is ______.
$$(2)$$ The first question recorded in the masterpiece mentioned in $$(1)$$ by Liu Hui is: In the diagram shown, to measure the height $$AH$$ of a mountain peak on a sea island, two measuring poles $$BC$$ and $$DE$$, each 3 zhang high, are erected. The distance between the two poles $$BD$$ is 1000 steps, with points $$D$$, $$B$$, and $$H$$ aligned. Retreating 123 steps from point $$B$$ to point $$F$$, with the observer’s eye at ground level observing point $$A$$, points $$A$$, $$C$$, and $$F$$ also align. Retreating 127 steps from point $$D$$ to point $$G$$, with the observer’s eye at ground level observing point $$A$$, points $$A$$, $$E$$, and $$G$$ also align. Calculate the height $$AH$$ in zhang and the distance $$HB$$ in steps $$($$where 1 step$$=6$$ feet, and 1 zhang$$=10$$ feet$$)$$. | 753 |
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<image>As shown in the , in triangle ABC, D, E, and F are points on sides BC, BA, and AC respectively, ∠B = ∠DEB, ∠C = ∠DFC. If ∠A = 70°, find the degree measure of ∠EDF. | 40 |
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<image>ABC takes on a sector shape. EDF takes the form of a sector. How can we calculate the perimeter of EDF? | 11*π/6 + 22 |
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<image>As shown in the figure, in the parallelogram $$ABCD$$, the bisector of $$∠BAD$$, $$AE$$, intersects side $$CD$$ at point $$E$$. Given that $$AB=5cm$$ and $$BC=3cm$$, find the length of $$EC$$ in centimeters. | 2 |
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<image>ABC is in the form of a sector. DEF describes an equilateral triangle. Square FDGH contains an inscribed circle. What is the perimeter of circle? | 60*π |
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<image>As shown in the figure, given \( AB \parallel CD \), \(\angle C = 35^\circ \), and \( BC \) bisects \(\angle ABE \), then the degree measure of \(\angle ABE\) is ______(degrees). | 70° |
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<image>$\textbf{Bake Sale}$
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
<image1>
$\circ$ Roger's cookies are rectangles:
<image2>
$\circ$ Paul's cookies are parallelograms:
<image3>
$\circ$ Trisha's cookies are triangles:
<image4>
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? | 40 |
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<image>As shown in the figure, a square with a side length of $$3a$$ is cut along the dashed lines into two smaller squares and two rectangles. If the smaller square with a side length of $$2b$$ is removed, and the remaining three pieces are assembled into a rectangle, then the length of the longer side of this rectangle is ______. | 3a + 2b |
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<image>ABC takes on a sector shape. CBDE is a parallelogram. EDFG conforms a parallelogram. Square GFHI includes an inscribed circle. Can you calculate the area of the entire inscribed circle? | π |
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<image>(2013 Autumn • Bijie region school-level end-of-term) As shown in the figure, C and D are two points on the line segment AB. If CB = 4 cm, DB = 7 cm, and D is the midpoint of AC, then the length of the line segment AB is ____. | 10 |
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<image>ABCD conforms a square. DCE is an isosceles triangle, having DC and CE of the same length. Square ECFG contains an inscribed circle. How long is EG in shape ECFG? | 3 |
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<image>CDE constitutes an equilateral triangle. GFH is geometrically a sector. Can you tell me the perimeter of GFH? | π + 4 |
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<image>(2015 Spring • Panji District Midterm) As shown in the figure, triangle ABC with a perimeter of 18 is translated 2 units to the right to obtain triangle DEF. The perimeter of quadrilateral ABFD is ____. | 22 |
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<image>According to the image, it is known that points D and E are on sides AB and AC of △ABC, respectively. DE is parallel to BC, and the area of △ADE to the area of quadrilateral DBCE is 1:3. Therefore, AD:AB = ___ .
| 1:2 |
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<image> As shown in the figure, in \(\triangle ABC\), it is known that \(DE \parallel BC\), and \(\dfrac{AE}{EC} = \dfrac{2}{3}\). Then, the ratio of the area of \(\triangle ADE\) to the area of \(\triangle ABC\) is ______. | 4:25 |
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<image> As shown in the figure, $$ABCD$$ is a cyclic quadrilateral of $$⊙O$$, point $$E$$ is on the extension of $$AB$$, $$BF$$ is the bisector of $$∠CBE$$, and $$∠ADC=110^{\circ}$$. Then $$∠FBE$$ equals ______. | 55^\circ |
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<image> As shown in the figure, it is a plane unfolding of a cube, with a Chinese character on each face. On the surface of the original cube, the character opposite to the Chinese character “香” is ___. | 泉 |
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<image>(Spring 2014 • End of term in Yancheng district) As shown in the figure, in parallelogram ABCD, CE bisects ∠DCB, F is the midpoint of AB, AB = 8, BC = 6, then AE:EF:FB = ____. | 1:1:2 |
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<image>As shown in the figure, $$AB$$ is the diameter of the circle $$⊙O$$, chords $$AD$$ and $$BC$$ intersect at point $$P$$, and $$AD=BC$$.
$$(1)$$ Prove: $$\triangle ACB$$ ≌ $$\triangle BDA$$;
$$(2)$$ If $$∠ABC=35^{\circ}$$, then $$∠CAP=$$ ______ $$ {\,\!}^{\circ}.$$ | \triangle ACB≌\triangle BDA |
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<image>Find the area of the shaded region in the following figure. (Unit: m)
| 24 |
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<image>As shown in the figure, AB ∥ CD, AD and BC intersect at point O, ∠A = 20°, ∠COD = 100°, then the measure of ∠C is (____)
| 60 |
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<image>As shown in the figure, in △ABC, D is a point on the extension of AB, ∠A = 40°, ∠C = 60°, then ∠CBD = ______ . | 100 |
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<image>The figure shown is composed of four squares with equal side lengths. The degree measure of ∠ABC in the figure is __ ▲_ ._ | 45 |
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<image> As shown in the figure, a circle with a radius of $$1$$ unit rolls one full revolution to the right along the number line from the origin. A point $$A($$ which coincides with the origin when rolling starts $$)$$ on the circle reaches point $$B$$. Therefore, the length of $$AB$$ is equal to the circumference of the circle, ______, so the number represented by point $$B$$ on the number line is ______, which is a ______ number. | 2\pi |
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<image>The vertices of a $ 3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
A. $12\pi$
B. $\frac{25\pi}{2}$
C. $13\pi$
D. $\frac{27\pi}{2}$
E. $14\pi$ | E. $14\pi$ |
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<image>As shown in the figure, $$D$$ is a point on the hypotenuse $$BC$$ of the right triangle $$\triangle ABC$$, and $$AC = \sqrt{3} DC$$.
$$(1)$$ If $$∠DAC = \dfrac{π}{6}$$, find the measure of angle $$B$$;
$$(2)$$ If $$BD = 2DC$$ and $$AD = 2$$, find the length of $$DC$$. | 60^\circ |
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<image>As shown in the figure, D and E are points on AB and AC respectively. If ∠A = 70°, ∠B = 60°, and DE ∥ BC, then the measure of ∠AED is ___ degrees.
| 50 |
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