How to use the Intersecting chord theorem

A chord is a line segment that has both of its endpoints on the edge of the circle

A chord of a circle is a line segment that has both of its endpoints on the circumference of a circle.

$\overline{BC}$ is an example of a chord.

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Intersecting chord theorem

The intersecting chord theorem states that the products of chord segments are always equal. For instance, consider chords $\overline{BD}$ and $\overline{EF}$,

then the intersecting chord theorem says that

$BC\cdot CD=EC\cdot CF$

Solving for chord length using the intersecting chord theorem

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Apply the intersecting chord theorem to find the length of the chords

Example

Find the value of $x$ in the figure.

The products of the chord segments are equal. So we can set up an equation.

$5(5x+5)=5(3x+12)$

$25x+25=15x+60$

$10x=35$

$x=3.5$

The intersecting chord theorem states that the products of chord segments are always equal.

Example

Find the length of the each chord.

First we need to find the value of $x$, and then use that to find the length of the chords. The products of the chord segments are equal, so

$12(x-4)=9(x-2)$

$12x-48=9x-18$

$3x=30$

$x=10$

Now we can find the length of each chord. One chord has a length of

$12+x-4$

$12+10-4$

$18$

The other chord has a length of

$x-2+9$

$10-2+9$

$17$

The chords have lengths of $17$ and $18$.

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