Two circles of radii 6 cm and 6 cm intersect each other. The distance between their centers is 8 cm. What is the length of their common chord?
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If a geometry problem's given numbers don't lead to any of the multiple-choice answers, check for a simple typo that might make the problem symmetric or easier. Assuming equal radii is a common correction.
Step 1: In the right triangle formed by a centre, the midpoint of the chord, and an intersection point, the half-distance between centres is 4 and the radius is 6, so the angle \( \theta \) between the centre-line and the radius satisfies \( \cos\theta = \frac{4}{6} = \frac{2}{3} \). Step 2: Then \( \sin\theta = \sqrt{1-\frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \). Step 3: Half the chord equals radius times \( \sin\theta \): \[ 6\times\frac{\sqrt5}{3} = 2\sqrt5. \] Step 4: Double it for the full chord: \[ \boxed{4\sqrt{5}\text{ cm}} \]