Question

In the given figure, AC || BH and AD || EF. Show area (ΔAHF) = area (pentagon ABCDE). [Hint: Join BC and DE.]

[Hint: Join BC and DE.]

Answer 1

Join BC and DE.

From the given figure,

ar(ΔAHF) = ar(ΔACD) + ar(ΔAHC) + ar(ΔADF)

Also, ar(pentagon ABCDE) = ar(ΔACD) + ar(ΔABC) + ar(ΔAED)

It is given that, AC || BH and AD || EF

Since the area of triangles on the same base and between the same parallels are equal.

So, we can say

ar(ΔABC) = ar(ΔAHC) and ar(ΔAED) = ar(ΔADF)

Now we can write,  ar(ABCDE) = ar(ΔACD) + ar(ΔAHC) = ar(ΔADF)

Both ar(ΔAHF) and ar(pentagon ABCDE) are the same.

Hence, area area of ΔAHF = area of pentagon ABCDE