# Segments Formed by Two Intersecting Chords

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

## Formula

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

(purple), (dark purple): segments from a chord

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

## Proof

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

[*x*, 4] and [3, 8]

4⋅*x* = 3⋅8

## Example 2

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 *x*⋅*x* = 2⋅(3 + 5).