Question

In $\Delta$ABC, $\angle A$ is obtuse, PB$\perp$AC and QC$\perp$AB. Prove that $AB\times AQ=AC\times AP$.

Answer 1

Given in $△ABC$

$\angle A=$obtuse

$PB\perp AC$; $QC\perp AB$

Consider $△BPA$ and $△AQC$

$\angle P=\angle Q={90}^o$

$\angle PAB=\angle QAC$ (Vertically opposite angle)

$\Rightarrow$ $△BPA\sim △AQC$ (AAA similarity)

$\therefore$ $\frac{AB}{AC}=\frac{AP}{AQ}$

$\Rightarrow$ $AB\times AQ=AP\times AC$

Hence proved