Question

In $△LMN,PQ\parallel MN$ such that $LP=2cm$ and $PM=6cm$. Also $MN=20cm$. Calculate the ratio of $ar\left(PQNM\right)$ and $ar\left(LPQ\right)$.

• A. $12:1$
• B. $13:1$
• C. $14:1$
• D. $15:1$

Answer 1

The correct option is D $15:1$

In $△$LPQ and $△$LMN

$\angle$L $\cong$$\angle$L [Common angle]

$\angle$LPQ $\cong$$\angle$LMN [Corresponding angles]

$\therefore$ $△$LPQ $\cong$ $△$LMN [By AA test of similarity]

$\frac{ar\left(LPQ\right)}{ar\left(LMN\right)}$ = $\frac{L{P}^2}{L{M}^2}$ [Areas of similar triangles]

$\therefore$$\frac{ar\left(LPQ\right)}{ar\left(LMN\right)}$ = $\frac{2^2}{8^2}$

$\therefore$$\frac{ar\left(LPQ\right)}{ar\left(LMN\right)}$ = $\frac{4}{64}$

$\therefore$$\frac{ar\left(LPQ\right)}{ar\left(LMN\right)}$ = $\frac{1}{16}$

$\therefore$$\frac{ar\left(LMN\right)}{ar\left(LPQ\right)}$ = $\frac{16}{1}$ [By Invertendo]

$\therefore$$\frac{ar\left(LMN\right)-ar\left(LPQ\right)}{ar\left(LPQ\right)}$= $\frac{16-1}{1}$ [By dividendo]

$\therefore$$\frac{ar\left(PQNM\right)}{ar\left(LPQ\right)}$ = $\frac{15}{1}$