Chords $\overline{PR}$ and $\overline{QS}$ of a circle are perpendicular. Find $RS$.

kittykat Mar 20, 2024

#1**0 **

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.

Boseo Mar 20, 2024