Question

In the figure, O is the centre of the circle with radius 5 cm. If tangent OP = 13 cm and OP intersects the circle at E. Find the perimeter of the $\Delta PCD$, where CD is the tangent to the circle at E.

Answer 1

OP = 13cm
OA = 5cm
Pythagoras theorem, $\Delta POA$,
$AP=\sqrt{{13}^2-5^2}=12cm$
CD is a target, So, $\angle CEO={90}^{\circ }$
$In\Delta POA,tan\theta =\frac{OA}{AP}=\frac{5}{12}$ - (1)
$For\Delta PEC,tan\theta =\frac{CE}{PE}$
PE = OP – OE = 13 cm – 5 cm = 8 cm
$tan\theta =\frac{CE}{8m}$
$\frac{5}{12}=\frac{CE}{8}$
$CE=\frac{40}{12}=\frac{10}{3}$
$Similarly,ED=\frac{10}{3};CE=AC=\left(\frac{10}{3}\right)$
Perimeter of PCD, PC + CD + DP
= (PA – CA) + CE + ED + (PB + BD)
= PA + PB = 26 cm