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Tangent and Radius

In the given ...

Question

In the given diagram PM and PN are the tangents from P to the circle. Find the area of quadrilateral OMPN if OM = r and PM = 2r.

r^{2}

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2 r^{2}

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4 r^{2}

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\frac{r^{2}}{2}

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Solution

The correct option is B $2 r^{2}$
Consider triangle OMP, $∠ O M P = 90^{\circ}$
Since, angle between radius and tangent is $90^{\circ}$
Area of $Δ O M P = \frac{1}{2} O M . M P = \frac{1}{2} r .2 r = r^{2}$

OM = ON = r; (Radii)
PM = PN = 2r; (length of tangents)

Area of quadrilateral OMPN is
$= a r e a o f (Δ O M P + Δ O N P$
$= r^{2} + r^{2} = 2 r^{2}$

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In the given diagram PM and PN are the tangents from P to the circle. Find the area of quadrilateral OMPN if OM = r and PM = 2r.

The correct option is B 2r2Consider triangle OMP, ∠OMP=90∘ Since, angle between radius and tangent is 90∘ Area of ΔOMP=12OM.MP=12r.2r=r2 OM = ON = r; (Radii) PM = PN = 2r; (length of tangents) Area of quadrilateral OMPN is =area of(ΔOMP+ΔONP =r2+r2=2r2