If the angle of intersection at a point where the two circles with radii 5cm and 12cm inte…

Question

If the angle of intersection at a point where the two circles with radii $5$cm and $12$cm intersect is $90^\circ$, then the length (in cm) of their common chord is:

Accepted Answer

Orthogonal intersection makes $\triangle PC_1C_2$ right-angled at $P$ (intersection point), with legs $5,12$ and hypotenuse $C_1C_2=13$.

The common chord is perpendicular to $C_1C_2$; the distance from $P$ to $C_1C_2$ (the altitude to the hypotenuse) is $\dfrac{5\times12}{13}=\dfrac{60}{13}$, and this equals half the common chord:

$$\text{Common chord}=2\times\frac{60}{13}=\frac{120}{13}.$$

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