Draw a circle of radius $6\ cm$. Take a point $P$ at a distance of $10\ cm$ from the centre $O$ of the circle. Draw a tangent $PT$ to the circle from point $P$. Draw a random secant $P-A-B$ to the circle. Then $\dfrac{PA \times PB}{PT} = $ ?

Using the tangent-secant theorem, we know that PT squared equals PA times PB. We can find PT by applying the Pythagorean theorem to the right triangle formed by the radius OT, the tangent PT, and the line OP, yielding PT squared equals 10 squared minus 6 squared. Thus, PT squared equals 64, so PT is 8 cm. The expression (PA times PB) divided by PT becomes PT squared divided by PT, which simply equals PT. Therefore, the result is 8.

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