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Home » Circles » Chords » Finding the Length of a Common Chord

Finding the Length of a Common Chord

Last updated: Jul 20, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

If we know the radii of two intersecting circles, and how far apart their centers are, we can calculate the length of the common chord.

Problem

Circles O and Q intersect at points A and B. The radius of circle O is 16, and the radius of circle Q is 9. Line OQ connects the centers of the two circles and is 20 units long. Find the length of the common chord AB.

common chord

Strategy

We know that line OQ is the perpendicular bisector of the common chord AB. And we are also given the lengths of the radii, so we probably need to use that. Let's draw these radii:

circles with radii

Now, we have two triangles, △AOC and △AQC. They are both right triangles (since OQ is perpendicular to AB), and both have the same height, h. If the base of one of these is x units long, the other base is 20-x, as OQ is 20 units long.

Now, using the Pythagorean Theorem and some basic algebra, we can solve the following system of equations for x:

(1) h2+(20-x)2=162 (in right triangle △AOC)
(2) h2+x2=92 (in right triangle △AQC)

And once we find x, we substitute it in one of the above equations to find h. AB is 2h, since OQ is a bisector of AB.

Solution

(1) AB⊥OQ //Line connecting centers is perpendicular to common chord
(2) △AOC, △AQC are right triangles
(3) h2+(20-x)2=162 //(2), Pythagorean theorem
(4) h2+x2=92 //(2), Pythagorean theorem
(5) 400-40x = 256-81 //Subtract (4) from (3)
(6) x=225/40=5.625
(7) h2=81-31.64
(8) h=√49.36=7.03
(9) AB=14.06

« The Diagonals of a Kite are Perpendicular to Each Other
The Axis of Symmetry of a Deltoid Bisects the Other Diagonal »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
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    • Pythagorean Theorem
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    • Similar Triangles
    • Triangle Inequalities

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