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

Given: In a triangle ABC, E is the midpoint of median AD.

To space Prove space colon space ar left parenthesis increment BED right parenthesis equals 1 fourth ar left parenthesis increment ABC right parenthesis.

Proof: In ΔABC,

Given: In a triangle ABC, E is the midpoint of median AD. Proof: In

∵ AD is a median.

therefore space space ar left parenthesis increment ABD right parenthesis equals ar left parenthesis increment ACD right parenthesis equals 1 half ar left parenthesis increment ABC right parenthesis

(1)

∵ A median of a triangle divides it into two triangles of equal areas.
In ΔABD,
∵ BE is a median.

therefore space space ar left parenthesis increment BED right parenthesis equals ar left parenthesis increment BEA right parenthesis equals 1 half ar left parenthesis increment ABD right parenthesis

∵ A median of a triangle divides it into two triangles of equal areas

rightwards double arrow space space ar left parenthesis increment BED right parenthesis equals 1 half ar left parenthesis increment ABD right parenthesis space space space space space space space space space space space space space space equals 1 half.1 half space ar left parenthesis increment ABC right parenthesis space space space space space space space space space space space space space vertical line space From space left parenthesis 1 right parenthesis space space space space space space space space space space space space space space equals space 1 fourth ar space left parenthesis increment ABC 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

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