Question

In the figure, $ABC$ and $AMP$ are two right triangles, right angled at $B$ and $M$ respectively, prove that:

(i) $△ABC~△AMP$

(ii)$\frac{CA}{PA}=\frac{BC}{MP}$

Answer 1

Step 1: To prove that the given triangles are congruent

It is given that $ABC$and $AMP$are two right triangles, right angled at $B$ and $M$ respectively

In $△ABC$and $△AMP$

$\angle ABC=\angle AMP$ ($\because$ both the angles are equal to $90°$)

$\angle BAC=\angle PAM$ ( Each $=\angle A$)

$\Rightarrow △ABC~△AMP$ ( By AA similarity criteria)

Step 2: To prove the condition $\frac{CA}{PA}=\frac{BC}{MP}$

As $△ABC~△AMP$ and corresponding sides of similar triangle are in same ratio.

$\Rightarrow \frac{CA}{PA}=\frac{BC}{MP}$

Hence it is proved that $△ABC~△AMP$ and $\frac{CA}{MP}=\frac{BC}{MP}$