Question

In a circle of radius 13 cm, a chord is at a distance of 12 cm from the center of the circle. Find the length (in cm) of the chord.

• A. 5 cm
• B. 10 cm
• C. 12 cm
• D. 8 cm

Hint:

The perpendicular from the center of the circle to a chord bisects the chord, use Pythagoras theorem to find chord length.

Answer 1

The Correct Option is B

Let the circle have center \(O\), radius \(r = 13 \text{ cm}\), and chord \(AB\) at a distance \(d = 12 \text{ cm}\) from the center.
Draw a perpendicular from \(O\) to chord \(AB\), meeting at point \(M\). Then, \(OM = 12 \text{ cm}\) and \(AM = MB = \frac{AB}{2}\).
Using the right triangle \(OAM\), by Pythagoras theorem:
\[ OA^2 = OM^2 + AM^2 \] \[ 13^2 = 12^2 + AM^2 \] \[ 169 = 144 + AM^2 \] \[ AM^2 = 169 - 144 = 25 \] \[ AM = 5 \text{ cm} \] Length of chord \(AB = 2 \times AM = 2 \times 5 = 10 \text{ cm}\).