0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What if you had circles of different radii? $\endgroup$ – geo_freak Sep 15 '17 at 0:11
  • $\begingroup$ 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. $\endgroup$ – Mathemagical Sep 15 '17 at 0:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.