Question

PQ is a chord of length $8$ cm of a circle of radius $5$ cm. The tangents at P and Q intersects at a point T. Then the length of TP is

• A. $10\frac{2}{7}$ cm
• B. $24.5$ cm
• C. $\frac{20}{3}$ cm
• D. $12$ cm

Answer 1

The correct option is C $\frac{20}{3}$ cm

Let $2\theta$ be the angle of tangent

Given, $PO=OQ=5cmQP=8cm$

Let $M$ be the midpoint of $PQ$

$PM=MQ=4cm$

Also, $\angle OPQ=\theta \angle OPT=\angle OQT={90}^o\angle OMP=\angle OMQ={90}^o$

In $△OMP,\cos \theta =\frac{4}{5},then\sin \theta =\frac{3}{5},tan\theta =\frac{3}{4}$

In $△OPT,\tan \theta =\frac{OP}{PT}=\frac{5}{PT}PT=\frac{5}{\tan \theta }=\frac{5}{3}\times 4=\frac{20}{3}cm$