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?
Best Answer
By symmetry - because both circles are of same radius. Otherwise, you're right, there is no reason for that leg of the triangle to be half the distance between centers.