Animated Solution for Mathematics - Circles: If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is :

Imagine two circles with radii $r_1=5\text{ cm}$ and $r_2=12\text{ cm}$. They intersect at two points, and we are given that they intersect orthogonally.

This condition implies that the tangents at the point of intersection are perpendicular. Consequently, the radii drawn to the point of intersection are also perpendicular to each other.

Let the centers of the circles be $C_1$ and $C_2$, and let $P$ be one of the points of intersection. We have formed a right-angled triangle $△C_{1}P{C}_2$ with legs of length $5\text{ cm}$ and $12\text{ cm}$.

By the Pythagorean theorem, the distance between the centers $d$ is:

The line connecting the centers is the perpendicular bisector of the common chord. Let the chord intersect the line of centers at point $M$. The segment $PM$ represents the altitude of the right-angled triangle $△C_{1}P{C}_2$ dropped onto the hypotenuse $C_1{C}_2$.

We can determine the length of $PM$ by equating two different expressions for the area of the triangle:

Equating these, we find:

Since the line of centers bisects the common chord, the total length of the chord is twice the length of $PM$.

The final length of the common chord is $\frac{120}{13}\text{ cm}$.