Question

In the given figure, the diameter CD of a circle with centre O is perpendicular to chord AB. If AB = 12 cm and CE = 3 cm, calculate the radius of the circle.

Answer 1

CD is the diameter of the circle with centre O and is perpendicular to chord AB.
Join OA.

Given: AB = 12 cm and CE = 3 cm
Let OA = OC = r cm (Radii of a circle)
Then OE = (r - 3) cm
Since the perpendicular from the centre of the circle to a chord bisects the chord, we have:
$AE\mathit{=}\left(\frac{\mathit{A}\mathit{B}}{\mathit{2}}\right)=\left(\frac{12}{2}\right)\mathrm{cm}=6\mathrm{cm}$
Now, in right angled ΔOEA, we have:
⇒ OA² = OE² + AE²
⇒ r² = (r − 3)² + 6²
⇒ r² = r² − 6r + 9 + 36
⇒ r² − r² + 6r = 45
⇒ 6r = 45
$\Rightarrow r=\left(\frac{45}{6}\right)\mathrm{cm}=7.5\mathrm{cm}$
⇒ r = 7.5 cm
Hence, the required radius of the circle is 7.5 cm.