Segments Formed by Two Intersecting Chords

Segments Formed by Two Intersecting Chords

How to solve the segments formed by two intersecting chords problems: formula, examples, and their solutions.

Formula

[blue]*[dark blue] = [green]*[dark green]. [blue], [dark blue]: Segments from a chord. [green], [dark green]: Segments from the other chord.

[blue]⋅[dark blue] = [green]⋅[dark green]

[blue], [dark blue]: Segments from a chord
[green], [dark green]: Segments from the other chord

Example 1

Find the value of x. Segments formed by two intersecting chords: x, 4, 3, 8.

Four segments are formed
by two intersecting chords.

The segments are [x, 4] and [3, 8].

So x⋅4 = 3⋅8.

Divide both sides by 4.

Then x = 3⋅2.

3⋅2 = 6

So x = 6.

Example 2

Find AB. OP = 3, OC = 5.

Previously, you've solved this problem.

Chord of a circle - Example

Let's solve the same example
by using the formula you've just learned.

To use the formula,
make the other chord.

The radius is 5.
And OP = 3.

So the green segment is, 5 - 3, 2.

Recall that
if a segment starts from the radius
and is perpendicular to the chord,
then the segment bisects the chord.

Chord of a circle

OP is perpendicular to AB,
which is the chord of the given circle.

So OP bisects AB.

So set AP = PB = x.

Four segments are formed
by two intersecting chords.

The segments are [x, x] and [2, 8].

So xx = 2⋅8.

xx = x2
2⋅8 = 16

Square root both sides.

Then x = √16

x > 0
So you don't have to write (±)
in front of the square root sign.

16 = 42

Then cancel the square and the square root.

Then √42 = 4.

Square root

AB = 2x

x = 4

So AB = 2⋅4.

2⋅4 = 8

So AB = 8.

As you can see,
you can get the same answer.