Question

$\Delta LMN\sim \Delta PQR$, $9Ar\left(\Delta PQR\right)=16Ar\left(\Delta LMN\right)$. If $QR=20$ then Find $MN$.

Answer 1

Given:

$\Delta LMN\sim \Delta PQR$

$9Ar\left(\Delta PQR\right)=16Ar\left(\Delta LMN\right)$

$\frac{Ar\left(\Delta PQR\right)}{Ar\left(\Delta LMN\right)}=\frac{16}{9}$ ---(1)

$QR=20$---(given)

If two triangles are similar then, the ratio of the areas of two triangles is equal to squares the ratio of their corresponding sides.

Since, $\Delta PQR\sim \Delta LMN$

$\Rightarrow \frac{Ar\left(\Delta PQR\right)}{Ar\left(\Delta LMN\right)}=\frac{Q{R}^2}{M{N}^2}$

$\Rightarrow \frac{16}{9}=\frac{{20}^2}{M{N}^2}$

$\Rightarrow M{N}^2\times 16={20}^2\times 9$

$\Rightarrow M{N}^2\times 16=400\times 9$

$\Rightarrow M{N}^2=\frac{400\times 9}{16}$

$\Rightarrow M{N}^2=25\times 9$

$\Rightarrow MN=5\times 3$

$\therefore MN=15$.