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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.

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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.

Let ABC and A‘BC be two triangles on the same base BC and between the same parallels PQ and RS.

Draw BM ⊥ PQ. Then,

ar left parenthesis increment ABC right parenthesis equals 1 half cross times Base space cross times space Corresponding space attitude space space space space space space space space space space space space space equals 1 half cross times BC cross times BM space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis ar left parenthesis increment straight A apostrophe BC right parenthesis equals 1 half cross times space Base space cross times space Corresponding space attitude space space space space space space space space space space space space space equals 1 half cross times BC cross times BM space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

From (1) and (2), we get
ar(AABC) = ar(ΔA'BC)
Hence, ΔABC and ΔA'BC are equal in area.

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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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