Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $ riangle PQR$. Show that $ riangle ABC \sim riangle PQR$.

Question

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $	riangle PQR$. Show that $	riangle ABC \sim 	riangle PQR$.

Accepted Answer

Step 1: Triangles ABC and PQR with medians AD and PM drawn to sides BC and QR respectively
Step 2: In $	riangle ABC$ and $	riangle PQR$: $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM}$. $\frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM}$ ($\because$ D is midpoint of BC and M is midpoint of QR)
Step 3: $\frac{AB}{PQ} = \frac{BD}{QM} = \frac{AD}{PM} \Rightarrow 	riangle ABD \sim 	riangle PQM$ (SSS similarity)
Step 4: $\Rightarrow \angle B = \angle Q$ --- (i)
Step 5: Now, in $	riangle ABC$ and $	riangle PQR$: $\frac{AB}{PQ} = \frac{BC}{QR}$ (given) and $\angle B = \angle Q$ from (i)
Step 6: $	herefore 	riangle ABC \sim 	riangle PQR$ (SAS similarity)

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