Sides AB and BC and the median AD of a triangle AВС are respectively proportional to the sides PQ and QR and the…
CBSE Class 10 Maths PYQ · Triangles · Similarity with Triangles · 3 Marks · July 2023 · Standard
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943 Marks · July 2023 · Standard
Sides AB and BC and the median AD of a triangle AВС are respectively proportional to the sides PQ and QR and the median PM of $\triangle PQR$. Show that $\triangle ABC \sim \triangle PQR$.
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Given $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM}$ Since AD and PM are medians, $BC = 2BD$ and $QR = 2QM$ So, $\frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM} \Rightarrow \frac{AB}{PQ} = \frac{BD}{QM} = \frac{AD}{PM}$ Therefore, $\triangle ABD \sim \triangle PQM$ (SSS similarity criterion) $\Rightarrow \angle B = \angle Q$ (Corresponding angles of similar triangles) Now, in $\triangle ABC$ and $\triangle PQR$ $\frac{AB}{PQ} = \frac{BC}{QR}$ (Given) $\angle B = \angle Q$ (Proved above) Therefore, $\triangle ABC \sim \triangle PQR$ (SAS similarity criterion)
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