Question

In figure, QA and PB are perpendicular to AB. If AO = 10cm, BO = 6cm and PB = 9cm.
Find AQ.

Answer 1

In triangles AOQ and BOP, we have
$\angle OAQ=\angle OBP$ [Each equal to ${90}^o$]
$\angle AOQ=\angle BOP$ [Vertically opposite angles]
Therefore, by AA-criterion of similarity
$△AOQ\sim △BOP$
$\Rightarrow \frac{AO}{BO}=\frac{OQ}{OP}=\frac{AO}{BP}$
$\Rightarrow \frac{AO}{BO}=\frac{AQ}{BP}\Rightarrow \frac{10}{6}=\frac{AQ}{9}$
$\Rightarrow AQ=\frac{10\times 9}{6}=15cm$