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

**(a)** 12 cm

**(b)** 13 cm

**(c)** 8.5 cm

**(d)** $\sqrt{199}$ cm

Given:

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

To do:

We have to find the length of $PQ$.

Solution:

Radius of the circle $= 5\ cm$

$OQ = 12\ cm$

We know that,

The tangent to a circle is perpendicular to the radius through the point of contact.

$\angle OPQ = 90^o$

Therefore, by Pythagoras theorem,

$PQ^2+OP^2 = OQ^2$

$PQ^2=OQ^2-OP^2$

$PQ^2 = 12^2 - 5^2$

$= 144 - 25$

$= 119$

$PQ = \sqrt{119}\ cm$

Therefore, the length of $PQ$ is $\sqrt{119}\ cm$.

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