In Fig. $\displaystyle 4$, PQ is a chord of length $\displaystyle 8$ cm of a circle of radius $\displaystyle 5$ cm. The tangents at P and Q meet at a point T . Find the length of TP . Fig.

\[\begin{array}{c} O P=5 \mathrm{~cm}, P R=\frac{1}{2} P Q=4 \mathrm{~cm} \\ \therefore \triangle P R T \cong \triangle Q R T \\ \Rightarrow T R \perp P Q \\ \therefore O R=\sqrt{25-16}=3 \mathrm{~cm} \end{array} \]Let $\displaystyle T R=x \Rightarrow x^{2}+16=T P^{2}$Also $\displaystyle (3+x)^{2}-25=T P^{2}$Solving (i) and (ii) $\displaystyle x=\frac{16}{3} \mathrm{~cm}$ \[\therefore T P=\sqrt{\frac{256}{9}+16}=\frac{20}{3} \mathrm{~cm} \]