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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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