Question

A tangent $PQ$ at a point $P$ of a circle of radius $5cm$ meets a line through the centre $O$ at a point $Q$ so that $OQ=12cm$. Length of $PQ$ is

• A. $12cm$
• B. $13cm$
• C. $8.5cm$
• D. $\sqrt{119}cm$

Answer 1

Answer: (D)

$\sqrt{119}cm$

Given-

O is the center of a circle of radius 5cm,

PQ is a tangent to the given circle at P.

PQ meets the line OP passing through O & P.

$OQ=12$cm.

To find out- $PQ$

Solution-

We join OP.

Then OP is a radius of the given circle through the point of contact P of the tangent QP.

$\therefore OP\perp QP$

i.e $\angle OPQ={90}^o$ ....The angle between a tangent and the radius through the point of contact is ${90}^o$

$\therefore \Delta OPQ$ is a right one with OQ as the hypotenuse.

So, applying Pythagoras theorem, we have

$PQ=\sqrt{{OQ}^2-{OP}^2}=\sqrt{{12}^2-5^2}cm=\sqrt{119}$ cm.