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

Question

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 $∆ A B C ~ ∆ P Q R$ .

Smart Achievers · Accepted Answer

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

$∴$ $∆ A B C ~ ∆ P Q R$

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 $∆ A B C ~ ∆ P Q R$ .

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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 ∆ABC~∆PQR.

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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 $∆ A B C ~ ∆ P Q R$ .