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Areas of Parallelograms and Triangles

Short Answer Type

11. Parallelograms on the same base and between same parallels are equal in area. Prove this.

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Long Answer Type

12. In figure, AD is median of triangle ABC, E is the mid-point of AD and F is the mid-point of AE.
Prove that

ar open parentheses ABF close parentheses equals 1 over 8 ar left parenthesis ABC right parenthesis.

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Short Answer Type

13. ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

Given: ABCD is a quadrilateral and BD is one of its diagonals.
To Prove: ABCD is a parallelogram and to determine its area.
Proof: ∠ABD = ∠BDC (= 90°) | Given But these angles form a pair of equal alternate
interior angles for lines AB, DC and a transversal BD AB || DC
Also, AD = DC (= 3 cm) | Given
Hence, quadrilateral ABCD is a parallelogram. A quadrilateral is a parallelogram if its one pair of opposite sides are parallel and equal Now,
ar(||gm ABCD) = Base x corresponding altitude
= 3x4 = 12 cm²

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14. In the given figure, AB | | DC. Show that ar(BDE) = ar(ACED).

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15.

In the given figure, ABED is a parallelogram in which DE = EC. Show that area (ABF) = area (BEC)

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16. Areas of triangles on the same bases and between the same parallels are equal in. Prove it.

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17. Prove that the area of a trapezium is equal to half of the product of its height and sum of parallel sides.

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18.

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that
(i) ar(ABCD) = ar(EFCD)
(ii) ar(ABCD) = DC x AL.

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19. If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

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20. In the figure, PQRS is a parallelogram with PQ = 12 cm, altitudes corresponding to PQ and SP are respectively 8 cm and 10 cm. Find SP.

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