Question

Line $AB$ is $24$ metres in length and is tangent to the inner one of the two concentric circles at point $C$. Points $A$ and $B$ lie on the circumference of the outer circle. It is known that the radii of the two circles are integers. The radius of the outer circle is:

Accepted Answer

A. Let $R$ be the radius of the outer circle and $r$ be the radius of the inner circle. $C$ bisects $AB$, so $AC = 12$. In the right-angled triangle formed by the center $O$, point $C$, and point $A$: $$R^2 - r^2 = AC^2 = 12^2 = 144$$ $$(R - r)(R + r) = 144$$ Since $R$ and $r$ are positive integers, testing the Pythagorean triple $(5, 12, 13)$ yields: $$13^2 - 5^2 = 169 - 25 = 144$$ Thus, $R = 13$ metres.

Concentric Circles and Tangent Length

Explanation

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