Question

In figure, AB || PR and ar (ΔPAB) : ar (ΔPQR) = 1 : 2 Then PQ : AQ = ______.

Answer 1

In figure, AB || PR and ar (ΔPAB) : ar (ΔPQR) = 1 : 2 Then PQ : AQ = `underlinebb(sqrt(2) : sqrt(2) - 1)`.

Explanation:

Note: If AB were really parallel to PR the given area ratio 1 : 2 is impossible (for a segment inside a triangle the ratio `(ar(ΔPAB))/(ar(ΔPQR)) ≤ 1/4`. 

So the usual correction is that AB is parallel to QR; under that assumption triangles PAB and PQR are similar (AA).

So `((PA)/(PQ))^2 = (ar(ΔPAB))/(ar(ΔPQR)) = 1/2`

⇒ `(PA)/(PQ) = 1/sqrt(2)`

Then AQ = PQ – PA

= `PQ(1 - 1/sqrt(2))` 

Hence `(PQ)/(AQ) = 1/(1 − 1/sqrt(2))` 

= `sqrt(2)/(sqrt(2) - 1)` 

= `2 + sqrt(2)`

Therefore PQ : AQ = `2 + sqrt(2) : 1`.