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Circles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCircles1 Mark
Question
Write ‘True’ or ‘False’ and justify your answer.
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then $\text{OP}=\text{a}\sqrt{2}.$
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Answer

True.
Consider a tangent PT from an external point P on a circle with radius 'a'. OT and PT are radius and tangent respectively at contact point T. $\therefore\ \angle\text{T}=90^\circ$ As $\triangle\text{OPT}\cong\triangle\text{OPR}$ [By SSS criterion of congruence] $\therefore\ \angle\text{OPT}=\angle\text{OPR}=\frac{90^\circ}{2}=45^\circ$ $\therefore$ In right angle $\triangle\text{OPT},$ $\sin45^\circ=\frac{\text{OT}}{\text{OP}}$ $\Rightarrow\ \frac{1}{\sqrt{2}}=\frac{\text{a}}{\text{OP}}$ $\Rightarrow\ \text{OP}=\sqrt{2}\text{a}.$ Hence, the given statment is true.

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