If AD and PM are medians of triangles ABC and PQR, respectively where $ riangle ABC \sim riangle PQR$, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$.

Question

If AD and PM are medians of triangles ABC and PQR, respectively where $	riangle ABC \sim 	riangle PQR$, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$.

Accepted Answer

Step 1: Two similar triangles ABC and PQR with medians AD and PM respectively
Step 2: $	riangle ABC \sim 	riangle PQR$. AD and PM are medians of $	riangle ABC$ and $	riangle PQR$ respectively. $	herefore \frac{AB}{PQ} = \frac{BC}{QR}$
Step 3: $\frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{BD}{QM}$
Step 4: Also $\angle B = \angle Q$ ($	riangle ABC \sim 	riangle PQR$). $\Rightarrow 	riangle ABD \sim 	riangle PQM$ (SAS similarity)
Step 5: $\Rightarrow \frac{AB}{PQ} = \frac{AD}{PM}$. Hence proved.

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