CBSE2026Class Class 10 · Mathematics5 Marks · Long✅ Verified
(B) [OR] In the given figure, CM and RN are respectively the medians of $\triangle ABC$ and $\triangle PQR$. If $\triangle ABC \sim \triangle PQR$, then prove that: (i) $\triangle AMC \sim \triangle PNR$ (ii) $\triangle CMB \sim \triangle RNQ$
✅ Answer & Solution
Since $\triangle ABC \sim \triangle PQR$: $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}$ and $\angle A=\angle P, \angle B=\angle Q, \angle C=\angle R$ ...(i). Since CM and RN are medians: $AM=MB=\frac{1}{2}AB$ and $PN=NQ=\frac{1}{2}PQ$. (i) To prove $\triangle AMC \sim \triangle PNR$: From (i): $\frac{AB}{PQ}=\frac{CA}{RP} \Rightarrow \frac{2AM}{2PN}=\frac{CA}{RP} \Rightarrow \frac{AM}{PN}=\frac{CA}{RP}$. Also $\angle A = \angle P$ (common angle from similarity). By SAS similarity criterion: $\triangle AMC \sim \triangle PNR$ (Proved). (ii) To prove $\triangle CMB \sim \triangle RNQ$: Similarly, $\frac{BM}{QN}=\frac{BC}{QR}$ (using $BM=\frac{1}{2}AB$, $QN=\frac{1}{2}PQ$, and $\frac{AB}{PQ}=\frac{BC}{QR}$). Also $\angle B = \angle Q$. By SAS similarity criterion: $\triangle CMB \sim \triangle RNQ$ (Proved)
