The corresponding sides of $ riangle ABC$ and $ riangle PQR$ are in the ratio $3 : 5$. $AD \perp BC$ and $PS \perp QR$ as shown in the following figures:

(i) Prove that $ riangle ADC \sim riangl

Question

The corresponding sides of $	riangle ABC$ and $	riangle PQR$ are in the ratio $3 : 5$. $AD \perp BC$ and $PS \perp QR$ as shown in the following figures:

(i) Prove that $	riangle ADC \sim 	riangle PSR$
(ii) If $AD = 4$ cm, find the length of $PS$.
(iii) Using (ii) find $	ext{ar}(	riangle ABC) : 	ext{ar}(	riangle PQR)$

Accepted Answer

Step 1: As $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = \frac{3}{5}$
$\Rightarrow 	riangle ABC \sim 	riangle PQR$
$\Rightarrow \angle C = \angle R$
Step 2: (i) In $	riangle ADC$ and $	riangle PSR$:
$\angle ADC = \angle PSR = 90°$
and $\angle C = \angle R$
$	herefore 	riangle ADC \sim 	riangle PSR$ (AA similarity)
Step 3: (ii) $\frac{AD}{PS} = \frac{AC}{PR} = \frac{3}{5}$
$\Rightarrow \frac{4}{PS} = \frac{3}{5}$
$\Rightarrow PS = \frac{20}{3}$ cm
Step 4: (iii) $\frac{	ext{ar}(	riangle ABC)}{	ext{ar}(	riangle PQR)} = \frac{\frac{1}{2} 	imes BC 	imes AD}{\frac{1}{2} 	imes QR 	imes PS} = \frac{3}{5} 	imes \frac{3}{5} = \frac{9}{25}$
$	herefore 	ext{ar}(	riangle ABC) : 	ext{ar}(	riangle PQR) = 9 : 25$

Question

Solution