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OR
In the given figure, $CM$ and $RN$ are respectively, the medians of $\Delta ABC$ and $\Delta PQR$. If $\Delta ABC \sim \Delta PQR$, prove that :
(i) $\Delta AMC \sim \Delta PNR$
(ii) $\angle BCM = \angle QRN$
(iii) $\Delta BMC \sim \Delta QNR$
In the given figure, $CM$ and $RN$ are respectively, the medians of $\Delta ABC$ and $\Delta PQR$. If $\Delta ABC \sim \Delta PQR$, prove that :
(i) $\Delta AMC \sim \Delta PNR$
(ii) $\angle BCM = \angle QRN$
(iii) $\Delta BMC \sim \Delta QNR$
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OR
(b) (i) $\Delta ABC \sim \Delta PQR \Rightarrow \frac{AB}{PQ} = \frac{AC}{PR}$
$\Rightarrow \frac{AC}{PR} = \frac{\frac{1}{2} AB}{\frac{1}{2} PQ} \Rightarrow \frac{AC}{PR} = \frac{AM}{PN}$
Also $\angle A = \angle P$
$\therefore \Delta AMC \sim \Delta PNR$ (by SAS similarity criterion)
(ii) $\Delta AMC \sim \Delta PNR$ (from part (i))
$\therefore \angle ACM = \angle PRN$
Also $\angle ACB = \angle PRQ$ ( as $\Delta ABC \sim \Delta PQR$)
$\therefore \angle ACB - \angle ACM = \angle PRQ - \angle PRN$
$\Rightarrow \angle BCM = \angle QRN$
(iii) $\Delta ABC \sim \Delta PQR \Rightarrow \frac{AB}{PQ} = \frac{BC}{QR}$
$\Rightarrow \frac{BC}{QR} = \frac{\frac{1}{2} AB}{\frac{1}{2} PQ} \Rightarrow \frac{BC}{QR} = \frac{BM}{QN}$
Also $\angle B = \angle Q$
$\therefore \Delta BMC \sim \Delta QNR$ (by SAS similarity criterion)
(b) (i) $\Delta ABC \sim \Delta PQR \Rightarrow \frac{AB}{PQ} = \frac{AC}{PR}$
$\Rightarrow \frac{AC}{PR} = \frac{\frac{1}{2} AB}{\frac{1}{2} PQ} \Rightarrow \frac{AC}{PR} = \frac{AM}{PN}$
Also $\angle A = \angle P$
$\therefore \Delta AMC \sim \Delta PNR$ (by SAS similarity criterion)
(ii) $\Delta AMC \sim \Delta PNR$ (from part (i))
$\therefore \angle ACM = \angle PRN$
Also $\angle ACB = \angle PRQ$ ( as $\Delta ABC \sim \Delta PQR$)
$\therefore \angle ACB - \angle ACM = \angle PRQ - \angle PRN$
$\Rightarrow \angle BCM = \angle QRN$
(iii) $\Delta ABC \sim \Delta PQR \Rightarrow \frac{AB}{PQ} = \frac{BC}{QR}$
$\Rightarrow \frac{BC}{QR} = \frac{\frac{1}{2} AB}{\frac{1}{2} PQ} \Rightarrow \frac{BC}{QR} = \frac{BM}{QN}$
Also $\angle B = \angle Q$
$\therefore \Delta BMC \sim \Delta QNR$ (by SAS similarity criterion)