Question

Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS.

Answer 1

It is given that the diagonals of the trapezium PQRS intersect each other at the point O. Also, PQ || RS and PQ = 3RS.

In ∆SOR and ∆QOP,

∠SOR = ∠QOP (Vertically opposite angles)

∠OSR = ∠OQP (Alternate angles)

∴ ∆SOR $~$ ∆QOP (AA Similarity)

$\Rightarrow \frac{\mathrm{ar}\left(∆\mathrm{POQ}\right)}{\mathrm{ar}\left(∆\mathrm{ROS}\right)}={\left(\frac{\mathrm{PQ}}{\mathrm{RS}}\right)}^2$ (The ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides)

$\Rightarrow \frac{\mathrm{ar}\left(∆\mathrm{POQ}\right)}{\mathrm{ar}\left(∆\mathrm{ROS}\right)}={\left(\frac{3\mathrm{RS}}{\mathrm{RS}}\right)}^2=\frac{9}{1}$

Hence, the ratio of areas of triangles POQ and ROS is 9 : 1.