Question

Find the length of the tangent drawn to a circle with radius 8 cm from a point 17 cm away from the centre of the circle.

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{O}}\text{be the centre of the given circle}\text{.} \\ \text{Let}\mathit{\text{P}}\text{be a point, such that} \\ \mathit{\text{OP}}=17\text{cm}\text{.} \\ \text{Let}\mathit{\text{OT}}\text{be the radius, where} \\ \mathit{\text{OT}}=5\text{cm} \\ \text{Join}\mathit{\text{TP}}\text{,where}\mathit{\text{TP}}\text{is a tangent}\text{.} \\ \text{Now,tangent drawn from an external point is perpendicular to the radius} \\ \text{at the point of contact}\text{.} \\ \therefore \text{OT}\perp \text{PT} \\ \text{In the right}△\mathit{\text{OTP}}\text{, we have:} \\ {\mathit{\text{OP}}}^2={\mathit{\text{OT}}}^2+{\mathit{\text{TP}}}^2\left[\text{By Pythagoras' theorem:}\right] \\ \mathit{\text{TP}}=\sqrt{{\mathit{\text{OP}}}^2-O{T}^2} \\ =\sqrt{{17}^2-8^2} \\ \text{=}\sqrt{289-64} \\ \text{=}\sqrt{225} \\ \text{=15 cm}\text{.} \\ \therefore \text{The length of the tangent is 15 cm}\text{.} \end{gathered}$$