Question

In the given figure, PA is perpendicular to AB; QB is perpendicular to AB and PA= QB. If PQ intersect AB at M, show that M is the midpoint of both AB and PQ.

Answer 1

Answer:

M is the midpoint of PQ

M is the midpoint of AB

Step-by-step explanation:

Given:

PA = QB ............(1)

In ΔAPM and ΔBQM,

∠M=∠M (common)

∠A=∠B=90

∠P=∠Q

ΔAPM and ΔBQM are equiangular

By AAA similarity,

ΔAPM and ΔBQM are similar.

Their corresponding sides are proportional.

$\frac{AP}{BQ}=\frac{PM}{QM}=\frac{AM}{BM}\\\\using\:(1)\\\\1=\frac{PM}{QM}=\frac{AM}{BM}\\\\\frac{PM}{QM}=1\\\\PM=QM$

This implies, M is the midpoint of PQ

$\frac{AM}{BM}=1\\\\AM=BM$

This implies, M is the midpoint of AB