Sides AB and AC and median AD to ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that

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Solution

Q.
Sides AB and AC and median AD to $∆$ ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that $∆ A B C ~ ∆ P Q R$ .

Ans.

Produce AD to E such that AD = DE and join EC

Produce PM to N such that PM = MN and join NR

$∆ A D B ≅ ∆ E D C$

$∴ A B = E C$

Similarly, $P Q = N R$

Since, $\frac{A B}{P Q} = \frac{A C}{P R} = \frac{A D}{P M}$

$\Rightarrow \frac{E C}{N R} = \frac{A C}{P R} = \frac{\frac{A E}{2}}{\frac{P N}{2}}$

$∴ ∆ A E C ~ ∆ P N R$

$\Rightarrow ∠ 1 = ∠ 2$

Similarly, $∠ 3 = ∠ 4$

Hence $∠ 1 + ∠ 3 = ∠ 2 + ∠ 4 \Rightarrow ∠ A = ∠ P$

Also, $\frac{A B}{P Q} = \frac{A C}{P R}$

$∴$ $∆ A B C ~ ∆ P Q R$

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