+1439 Chords $\overline{PR}$ and $\overline{QS}$ of a circle are perpendicular. Find $RS$.
+195 Since PR and QS are perpendicular chords, they bisect each other. This means that point O, the center of the circle, lies on the line segment connecting P and S.
We are given that PR=10 and QS=24. Since PR is bisected by O, segments PO and OR each have length 2PR=5. Similarly, segments QO and OS each have length 2QS=12.
Now, consider right triangle POS. We know the length of both legs, PO=5 and OS=12, so we can use the Pythagorean Theorem to find the length of the hypotenuse, which is segment RS.
By the Pythagorean Theorem:
RS2=PO2+OS2
RS2=52+122
RS2=25+144
RS2=169
Taking the square root of both sides (remembering that since we're dealing with lengths, we only care about the positive square root), we find:
RS=169
RS=13.
