Question

In the figure, $CD$ and $RS$ are respectively the medians of $△ABC$ and $△PQR$. If $△ABC\sim △PQR$, prove that
(i) $△ADC\sim △PSR$
(ii) $\frac{CD}{RS}=\frac{AB}{PQ}$

Answer 1

Given:

$CD$ is median of $\Delta ABC$

$RS$ is median of $\Delta PQR$

$\Delta ABC\sim \Delta PQR$

To prove that:

(i) $\Delta ADC\sim \Delta PSR$

(ii) $\frac{CD}{RS}=\frac{AB}{PQ}$

Proof:

(i)

$AD=\frac{AB}{2}$ as $CD$ is the median

$PS=\frac{PQ}{2}$ as $RS$ is the median

$\Delta ABC\sim \Delta PQR$

$⟹\frac{AB}{PQ}=\frac{AC}{PR}$

$\frac{2AD}{2PS}=\frac{AC}{PR}$

$⟹\Delta ADC\sim \Delta PSR$

Hence proved

(ii)

$\Delta ADC\sim \Delta PSR$

$⟹\frac{CD}{RS}=\frac{AB}{PQ}$

Hence proved