Question

If $△$ ABC and $△$ AMP are right triangles, right-angled at B and M respectively and $\angle MAP=\angle BAC$.

Prove that

(i) $△ABC\sim △AMP$

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

Answer 1

given that

$\angle ABC=\angle AMP=90$

and

$\angle MAP=\angle BAC$

therefore the other angle is also equal [since the sum of the angles must be 180]

since all the angles are equal the two triangles are similar

$△ABC\sim △AMP$

therefore, as the two triangles are similar, the ratio of sides is the same

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