Two circles of radii $r_{1}, r_{2}$ intersect orthogonally. The length of their common chord is
- $\dfrac {2r_{1}r_{2}}{\sqrt {r_{1}^{2} + r_{2}^{2}}}$
- $\dfrac {r_{1}r_{2}}{\sqrt {r_{1}^{2} + r_{2}^{2}}}$
- $\dfrac {2r_{1}^{2}r_{2}}{\sqrt {r_{1}^{2} + r_{2}^{2}}}$
- $\dfrac {2r_{2}^{2}r_{1}}{\sqrt {r_{1}^{2} + r_{2}^{2}}}$
For orthogonal circles, the distance between centers d^2 = r1^2 + r2^2. The length of the common chord is 2 * (r1 * r2) / d. Substituting d = sqrt(r1^2 + r2^2) gives 2 * r1 * r2 / sqrt(r1^2 + r2^2).
When two circles intersect orthogonally, the radii drawn to a point of intersection are perpendicular to each other, meaning the angle between the radius r1 and r2 at the intersection point is 90 degrees. Let the common chord have a length L; half of this chord, L/2, acts as the altitude to the hypotenuse in the right triangle formed by the two radii. In a right triangle, the altitude to the hypotenuse is the product of the legs divided by the hypotenuse, so L/2 = (r1 * r2) / sqrt(r1^2 + r2^2). Multiplying by 2 gives the length of the common chord as 2 r1 r2 / sqrt(r1^2 + r2^2).