Question

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

Answer 1

In triangles AOQ and BOP, we have

$$\angle OAQ=\angle OBP$$ [Each equal to $$90^\circ$$]

$$\angle AOQ=\angle BOP$$ [Vertically opposite angles]

Therefore, by AA-criterion of similarity

$$\triangle AOQ\sim \triangle BOP$$

$$\Rightarrow $$ $$\dfrac { AO }{ BO } =\dfrac { OQ }{ OP } =\dfrac { AQ }{ BP } $$

$$\Rightarrow $$ $$\dfrac { AO }{ BO } =\dfrac { AQ }{ BP } $$

$$\Rightarrow $$ $$\dfrac { 10 }{ 6 } =\dfrac { AQ }{ 9 } $$

$$\Rightarrow $$ $$AQ=\dfrac { 10\times 9 }{ 6 } =15 cm$$