Sides AB, BC and the median AD of △ ABC are respectively proportional to sides PQ, QR and the median PM of another △…

CBSE Class 10 Maths PYQ · Triangles · Similarity with Triangles · 5 Marks · March 2024 · Standard

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1125 Marks · March 2024 · Standard
Sides AB, BC and the median AD of $\triangle ABC$ are respectively proportional to sides PQ, QR and the median PM of another $\triangle PQR$. Prove that $\triangle ABC \sim \triangle PQR$.
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$\therefore \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM}$
$\therefore \frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM}$
$\Rightarrow \frac{AB}{PQ} = \frac{BD}{QM} = \frac{AD}{PM}$ ---------(i)
$\Rightarrow \triangle ABD \sim \triangle PQM$
$\Rightarrow \angle B = \angle Q$ ---------(ii)
In $\triangle ABC$ and $\triangle PQR$
$\frac{AB}{PQ} = \frac{BC}{QR}$
$\angle B = \angle Q$
$\therefore \triangle ABC \sim \triangle PQR$
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