Sides $AB$, $BC$ and the median $AD$ of $ riangle ABC$ are respectively proportional to sides $PQ$, $QR$ and the median $PM$ of $ riangle PQR$. Prove that $ riangle ABC \sim riangle PQR$.

Question

Sides $AB$, $BC$ and the median $AD$ of $	riangle ABC$ are respectively proportional to sides $PQ$, $QR$ and the median $PM$ of $	riangle PQR$. Prove that $	riangle ABC \sim 	riangle PQR$.

Accepted Answer

Step 1: Given $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM}$. Since $D$ is midpoint of $BC$ and $M$ is midpoint of $QR$: $\frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM}$
Step 2: $\Rightarrow 	riangle ABD \sim 	riangle PQM$ (SSS similarity)
Step 3: $\Rightarrow \angle B = \angle Q$
Step 4: In $	riangle ABC$ and $	riangle PQR$: $\frac{AB}{PQ} = \frac{BC}{QR}$ and $\angle B = \angle Q$
Step 5: $	herefore 	riangle ABC \sim 	riangle PQR$ (SAS similarity)

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