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Areas of Parallelograms and Triangles

Short Answer Type

31. In the figure, ABCD is a parallelogram. P is a point on AB produced and DN ⊥ AB. If AB = 8 cm and DN = 3 cm. Find the area of ΔCPD.

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32. ABCD is a parallelogram in which BC is produced to E such that CE = BC (see figure). AE intersects CD at F. Prove that ar(ΔBFC)

equals 1 fourth space ar left parenthesis ABCD right parenthesis.

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33. In the figure, AB || CD. E and F are the points on the sides CD and AB respectively. If ar(ΔABE) = ar(ΔDCF), then prove that AB = DC.

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34. In figure, E is any point on median AD of a ΔABC. Show that ar(ΔABE) = ar(ΔACE).

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35. In a triangle ABC, E is the midpoint of median AD. Show that

a r left parenthesis increment B E D right parenthesis space equals 1 fourth a r left parenthesis increment A B C right parenthesis

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36. Show that the diagonals of a parallelogram divide it into four triangles of equal area.

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37. In figure, ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O. Show that ar(AABC) = ar(ΔABD).

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Long Answer Type

38.

D, E and F are respectively the midpoints of the sides BC, CA and AB of a ΔABC. Show that:

(i) BDEF is a parallelogram

$left parenthesis ii right parenthesis space ar left parenthesis increment DEF right parenthesis equals 1 fourth ar left parenthesis increment ABC right parenthesis left parenthesis iii right parenthesis space ar left parenthesis square space space BDEF right parenthesis equals 1 half ar left parenthesis increment ABC right parenthesis$

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39.

In figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that:

(i) ar(ΔDOC) = ar(ΔAOB)
(ii) ar(ΔDCB) = ar(ΔACB)
(iii) DA || CB or ABCD is a parallelogram.
[Hint. From D and B, draw perpendiculars to AC.]

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Short Answer Type

40. D and E are points on sides AB and AC respectively of ΔABC such that ar(ΔDBC) = ar(ΔEBC). Prove that DE || BC.

Given: D and E are points on sides AB and AC respectively of ΔABC such that ar(ΔDBC) = ar(ΔEBC).

To Prove: DE || BC.
Proof: ∵ ΔDBC and ΔEBC are on the same base BC and have equal areas.
Their altitudes must be the same.
∴ DE || BC.

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