Question

A point P is at a distance of 29 cm from the centre of a circle of radius 20 cm. Find the length of the tangent drawn from P to the circle. [CBSE 2017]

Answer 1

Consider the figure.
We know that the tangent is perpendicular to the radius of a circle.
So, OPB is a right angled triangle, with $\angle \mathrm{OBP}=90°$
By using pythagoras theorem in $△\mathrm{OPB}$, we get
$$\begin{gathered} \Rightarrow {\mathrm{OB}}^2+{\mathrm{PB}}^2={\mathrm{OP}}^2 \\ \Rightarrow (20{)}^2+{\mathrm{PB}}^2=(29{)}^2 \\ \Rightarrow 400+{\mathrm{PB}}^2=841 \\ \Rightarrow {\mathrm{PB}}^2=841-400=441 \\ \Rightarrow \mathrm{PB}=\sqrt{441}=21 \end{gathered}$$
So, length of the tangent from point P is 21 cm.