Areas Of Parallelograms And Triangles

Asked by avimanyu | 18th Feb, 2010, 04:51: PM

Expert Answer:

KL is parallel to and half the measure of AC, because in triangle ABC
segment KL is a midsegment. 

The same can be said about MN.

So MN and KL are parallel and congruent. From this we see that KLMN is
a parallelogram.

Now let X and Y be the intersections of AC with KN and LM,
respectively. And let E be the foot of the altitude from B on AC.

KLYX is a parallelogram too. When KL is taken as base, then the
measure of the height of this parallelogram is half BE. We find:

Area KLYX = KL x 0.5 x BE
= 0.5x AC x 0.5 x BE
= 0.5 x (0.5 x AC xBE)
= 0.5 x Area ABC

In the same way we find that area XYMN = 0. 5 x area ACD.

Combining these two we find the desired result that:

Area KLMN = 0.5 x Area ABCD.
Try the other two on the similar lines. In case you do not get the solution
please post each query individually.


Answered by  | 19th Feb, 2010, 10:59: AM

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