Question

AD and PS are respectively, the medians of $△ A BC$ and $△ PQR$ . If $△ A BC \sim △ PQR$ , then prove that
(i) $△ A D C \sim △ PSR$
(ii) $\frac{A D}{PS} = \frac{BC}{QR}$

Approach

Since $△ A BC \sim △ PQR$ , corresponding sides are proportional and angles are equal. Medians AD and PS divide BC and QR in the ratio 1:1. Using SAS similarity, $△ A D C \sim △ PSR$ . From similarity, $\frac{A D}{PS} = \frac{D C}{SR} = \frac{BC /2}{QR /2} = \frac{BC}{QR}$ .

Step-by-step working

1
$△ A BC \sim △ PQR$ $\Rightarrow$ $\frac{A B}{PQ} = \frac{A C}{PR} = \frac{BC}{QR} = \frac{2 D C}{2 SR}$
2 marks
2
$\Rightarrow$ $\frac{A C}{PR} = \frac{D C}{SR}$ and $∠ C = ∠ R$
1 mark
3
$∴$ $△ A D C \sim △ PSR$ (by SAS similarity)
1 mark
4
Since $△ A D C \sim △ PSR$ , $\frac{A D}{PS} = \frac{D C}{SR} = \frac{BC /2}{QR /2} = \frac{BC}{QR}$
1 mark

Concepts used

similar trianglesmediansratio

Approach

Step-by-step working

Concepts used

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