When we have two chords that crossing inside a circle, as presented below, the two triangles that result are similar.

You are watching: A circle with two chords is shown below Figure \$$\\PageIndex1\$$

This renders the matching sides in each triangle proportional and leads to a relationship between the segment of the chords, as proclaimed in the Intersecting Chords Theorem.

Intersecting Chords Theorem: If two chords intersect inside a one so that one is separated into segment of length a and also b and the other into segments of length \$$c\$$ and also \$$d\$$ climate \$$ab=cd\$$. api/deki/files/2106/f-d_c0db90aef16124295c0a755d0ec27c53f981a10d60d5c9dc03424beb%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=332&height=332\" />Figure \$$\\PageIndex3\$$

Solution

Use the Intersecting Chords Theorem.

\$$15\\cdot 4=5\\cdot x\$$

\$$60=5x\$$

\$$x=12\$$

Example \$$\\PageIndex2\$$

Find \$$x\$$. Simplify any kind of radicals. api/deki/files/2108/f-d_5c60692f56058607f8028caa60f18b174d74bed0d8c8f3107209d32e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=312&height=311\" /> figure \$$\\PageIndex5\$$ api/deki/files/2110/f-d_b8462186de5521ca85b40aa2edc295a1e0ae69e9c2c0cbf34a06fafd%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=322&height=320\" /> figure \$$\\PageIndex7\$$ figure \$$\\PageIndex8\$$

Solution

use the Intersecting Chords Theorem.

\\\beginaligned 8\\cdot 24&=(3x+1)\\cdot 12 \\\\192&=36x+12 \\\\ 180&=36x \\\\ 5&=x\\endaligned\

use the Intersecting Chords Theorem.

\\\beginaligned (x−5)21&=(x−9)24 \\\\ 21x−105&=24x \\\\ 111&=3x \\\\ 37−216&=x \\endaligned\

## Vocabulary

Term definition
central angle An angle formed by two radii and also whose vertex is in ~ the center of the circle.
chord A heat segment who endpoints are on a circle.
circle The collection of every points that space the exact same distance far from a particular point, called the center.
diameter A chord the passes through the facility of the circle. The length of a diameter is 2 times the size of a radius.
inscribed angle An angle v its crest on the circle and whose sides room chords.
intercepted arc The arc that is within an inscriptions angle and also whose endpoints space on the angle.
radius The street from the center to the external rim that a circle.

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Intersecting Chords Theorem According to the Intersecting Chords Theorem, if 2 chords intersect inside a one so that one is divided into segment of length \$$a\$$ and also \$$b\$$ and the other into segments of length \$$c\$$ and also \$$d\$$, climate \$$ab = cd\$$.