# Find the Length of a Common Chord

The question is as follows:

Two circles of radius 10 cm are drawn so that their centers are 12 cm apart. The two points of intersection determine a common chord. Find the length of this chord.

I know that the line drawn from the radius to the chord will perpendicularly bisect the chord. I labeled half the length of the chord as the variable $x$. As I was trying to think of a possible measure of the other leg of the right triangle with the given answer that the total length of the chord is 16, I stumbled upon 6, which happened to be half of 12 (the distance of the two centers). And if I solve for $x$, then I get 8, which when multiplied by two, will give me 16cm. I want to know why the other leg of the right triangle was 6? Why did it need to be half of 12?

• Then these 3 sides form a triangle - the two radii and the line connecting the centers (length $b$). The half-chord is the length of an altitude of the triangle. It can even be calculated. $\frac{bh}{2}$ is the area of the triangle which can be obtained from Heron's formula, since you know all 3 sides. Then solve for $h$ and $2h$ is the length of your cord. – Mathemagical Sep 15 '17 at 0:47