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

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Question

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

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Accepted Answer

Step 1: $OP = 5$ cm, $PR = \frac{1}{2}PQ = 4$ cm. $	riangle PRT \cong 	riangle QRT \Rightarrow TR \perp PQ$. $OR = \sqrt{25 - 16} = 3$ cm.
Step 2: Let $TR = x$. Then $x^2 + 16 = TP^2$ ... (i)
Step 3: Also $(3 + x)^2 - 25 = TP^2$ ... (ii)
Step 4: Solving (i) and (ii): $x = \frac{16}{3}$ cm. $	herefore TP = \sqrt{\frac{256}{9} + 16} = \frac{20}{3}$ cm.

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