Sides $AB$ and $AC$ and median $AD$ of a $ riangle ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another $ riangle PQR$. Show that $ riangle ABC \sim riangle PQR$.

Question

Sides $AB$ and $AC$ and median $AD$ of a $	riangle ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another $	riangle PQR$. Show that $	riangle ABC \sim 	riangle PQR$.

Accepted Answer

Step 1: Produce $AD$ to $E$ such that $AD = DE$ and join $EC$. Produce $PM$ to $L$ such that $PM = ML$ and join $LR$.
Step 2: $	riangle ABD \cong 	riangle ECD$ (SAS) $\Rightarrow AB = EC$. Similarly, $PQ = LR$.
Step 3: $\frac{AB}{PQ} = \frac{AC}{PR} = \frac{AD}{PM} \Rightarrow \frac{EC}{LR} = \frac{AC}{PR} = \frac{2AD}{2PM} = \frac{AE}{PL}$
Step 4: $	herefore 	riangle AEC \sim 	riangle PLR \Rightarrow \angle 1 = \angle 4$. Similarly $\angle 1 = \angle 3$. Adding: $\angle BAC = \angle QPR$
Step 5: $	herefore 	riangle ABC \sim 	riangle PQR$ (SAS similarity)

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