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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).

Given: ABC and ABD are two triangles on the same base AB. Line segment CD is bisected by AB at O.
To Prove: ar(ΔABC) = ar(ΔABD).
Proof: ∵ Line segment CD is bisected by AB
at O.
∴ OC = OD BO is a median of ΔBCD and AO is a median of ΔACD
∵    BO is a median of ΔBCD
∴ ar(ΔOBC) = ar(ΔOBD)    ...(1)
∵    A median of a triangle divides it into two triangles of equal areas
∵    AO is a median of ΔACD
∴ ar(ΔOAC) = ar(ΔOAD)    ...(2)
∵    A median of a triangle divides it into two triangles of equal areas
Adding (1) and (2), we get ar(ΔOBC) + ar(ΔOAC)
= ar(ΔOBD) + ar(ΔOAD)
⇒ ar(ΔABC) = 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

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