Segments Formed by Two Intersecting Chords

Segments Formed by Two Intersecting Chords

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

Formula

If two chords intersect in a circle, then the product of the chord's segments is equal to the product of the other chord's segments.

(purple)⋅(dark purple) = (blue)⋅(dark blue)

(purple), (dark purple): segments from a chord
(blue), (dark blue): segments from the other chord

Proof

Segments Formed by Two Intersecting Chords: Proof of the Formula

Draw the green arc
and two additional white chords like above.

The red angles are the inscribed angle
of the green intercepted arc.
So the red angles are congruent.

The red dot angles are vertical angles.
So the red dot angles are also congruent.

See these two triangles.
These two have two pairs of congruent angles.

So, by the AA similarity,
these two triangles are similar triangles.

So the corresponding sides are proportional:
(ratio of top sides) = (ratio of bottom sides).

Similarity of sides in triangles

Then (purple)⋅(dark purple) = (blue)⋅(dark blue).

Proportion

Example 1

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

[x, 4] and [3, 8]

4⋅x = 3⋅8

Example 2

Find AB. OP = 3, OC = 5.

The example was in the previous page:
chord of a circle.
Let's solve this differently.

Draw the blue segment
to make the (blue)-(dark blue) chord.
The length of the blue segment is, 5 - 3, 2.

The (blue)-(dark blue) chord is perpendicular to AB.

Chord of a circle

Set the lengths of the bisected purple segments as x.

Then xx = 2⋅(3 + 5).