Question

In the given figure, AP = `1/2 PB`, then ar(ΔAPQ) : ar(ΔABC) is ______.

Options

• 1 : 9

• 9 : 1

• 1 : 4

• 4 : 1

Answer 1

In the given figure, AP = `1/2 PB`, then ar(ΔAPQ) : ar(ΔABC) is 1 : 9.

Explanation:

Since, AP = `1/2 PB` we get,

PB = 2AP

AB = AP + PB = 3AP

So, `(AP)/(AB) = 1/3`

Because PQ || BC, triangles △APQ and △ABC are similar.

Thus, the ratio of their areas is the square of the ratio of corresponding sides:

`"ar(ΔAPQ)"/"ar(ΔABC)" = ((AP)/(AB))^2`

= `(1/3)^2`

∴ `"ar(ΔAPQ)"/"ar(ΔABC)" = 1/9`

Hence, ar(ΔAPQ) : ar(ΔABC) = 1 : 9