Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.

Q.Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.

Two circles, each with a radius of 10 cm, have centers A and B and intersect at points P and Q.
PQ = 12 cm
AB intersects PQ at point O.
\(\therefore\) In \(\triangle\) AOP, AP = 10 cm
OP = \(\frac{PQ}{2}=\cfrac{12}{2}\) cm = 6 cm.
\(\therefore\) AO = \(\sqrt{AP^2 - OP^2}\)
\(=\sqrt{10^2 - 6^2}\) cm \(=\sqrt{100 - 36}\) cm \(=\sqrt{64}\) cm = 8 cm.
\(\therefore\) AB = 2AO = \(2 \times 8\) cm = 16 cm.
\(\therefore\) The distance between the centers of the two circles is 16 cm.