Question

Diagonals of a trapezium PQRS intersect each other at the point O. If PQ || RS and PQ = 3RS, then $\frac{ar\left(∆POQ\right)}{ar\left(∆ROS\right)}$= ________.

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 ∆ROS and ∆POQ,

∠SOR = ∠QOP (Vertically opposite angles)

∠OSR = ∠OQP (Alternate angles)

∴ ∆ROS $~$ ∆POQ (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)

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

Diagonals of a trapezium PQRS intersect each other at the point O. If PQ || RS and PQ = 3RS, then $\frac{\mathrm{ar}\left(∆\mathrm{POQ}\right)}{\mathrm{ar}\left(∆\mathrm{ROS}\right)}$ = ____9____.