$PQ$ is a chord of circle, which on producing, meets the tangent $TR$ at $R$. If $PQ = 5\ cm$ and $QR = 4 cm$, find TR.

Because R is an external point, the entire secant length is RQ plus QP, which equals 4 plus 5, or 9 cm. By the tangent-secant theorem, the square of the tangent TR equals the product of the external secant part and the entire secant length. Thus, TR squared equals 4 times 9, making TR equal to the square root of 36, which is 6 cm.

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