From a point P which is at a distance of 13 cm from the center O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then which of the following is the area of the quadrilateral PQOR?

A
${60 cm}^{2}$
B
${65 cm}^{2}$
C
${30 cm}^{2}$
D
${32.5 cm}^{2}$

Solution:

Given from a point P which is at a distance of 13 cm from the center O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn.
Constructing a diagram with the given data,

Here, in right angled triangle OPQ,

{OP}^{2} = {PQ}^{2} + {OQ}^{2}

{PQ}^{2} = {OP}^{2} - {OQ}^{2}

PQ = \sqrt{{OP}^{2} - {OQ}^{2}}

\Rightarrow \sqrt{13^{2} - 5^{2}}

\Rightarrow \sqrt{169 - 25}

\Rightarrow \sqrt{144}

\Rightarrow PQ = 12 cm

PQ = PR

[Two tangents from the same external point to a circle are equal].
Now,
Area of quadrilateral PQOR = 2 × Area of triangle POQ
Area of triangle =

\frac{1}{2} \times b \times h

Then,
Ar(PQOR) =

2 \times \frac{1}{2} \times 12 \times 5 {cm}^{2}

Ar(PQOR)

= 60 {cm}^{2}

Hence, the area of the required quadrilateral is

60 {cm}^{2}

.
Therefore, option 1 is correct.

From a point P which is at a distance of 13 cm from the center O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then which of the following is the area of the quadrilateral PQOR?

Solution:

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