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

 Mar 20, 2024
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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​.

 Mar 20, 2024

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