question_answer

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

A) \[\frac{20}{3}\,cm\]

B) 20 cm

C) 3 cm                            

D) 4 cm

Correct Answer: A

Solution :

Given, PQ = 8 cm, OP = 5 cm

Then, in \[\Delta OPR,\]

\[OR=\sqrt{O{{P}^{2}}-P{{R}^{2}}}=\sqrt{{{5}^{2}}-{{4}^{2}}}=3\,cm\]

Now, in similar \[\Delta TPO\] and \[\Delta PRO,\]

\[\frac{TP}{PO}=\frac{RP}{RO}\]\[\Rightarrow \]\[\frac{TP}{5}=\frac{4}{3}\]

\[\therefore \]      \[TP=\frac{20}{3}\,cm\]