The Angle Bisector Theorem
Simple Explanation
When a triangle's angle is bisected (cut into two equal angles) by a segment reaching the opposite side, that segment divides the opposite side into two pieces whose ratio equals the ratio of the two sides forming the bisected angle: BD/DC = AB/AC.
Why Do We Need It?
This theorem connects an angle-splitting construction to a length ratio, letting you find segment lengths from angle information (or vice versa) without measuring the angle directly.
See It
Triangle ABC with a bisector from vertex A splitting the angle at A into two equal parts and meeting side BC at point D
Formula
The Angle Bisector Theorem
If AD bisects ∠A in △ABC (D on BC), then BD/DC = AB/AC
The bisector of an angle of a triangle divides the opposite side into two segments whose ratio equals the ratio of the two sides forming that angle.
- D
- — the point where the angle bisector from A meets side BC
- BD, DC
- — the two segments the bisector divides BC into
- AB, AC
- — the two sides forming the bisected angle at A
When to use it: Whenever an angle bisector in a triangle is given, and you need to find how it divides the opposite side.
Worked Example
Apply the angle bisector theorem
In △ABC, AD bisects ∠A, with D on BC. If AB=6, AC=9, and BC=15, find BD and DC.
Why Does This Work?
Stated simply: draw a line through C parallel to AD, meeting the extension of BA at a point E. Since AD ∥ CE, alternate and corresponding angles show △AEC is isosceles with AE = AC. Then in △BCE, AD ∥ CE means BD/DC = BA/AE (basic proportionality theorem) — and since AE = AC, this is exactly BD/DC = AB/AC.
Real-Life Example
Dividing land fairly along a bisecting boundary
A triangular plot of land is divided by a straight boundary line from one corner that bisects that corner's angle.
The angle bisector theorem tells surveyors exactly how that boundary divides the opposite edge, in the same ratio as the two adjacent side lengths.
Practice
In △ABC, AD bisects ∠A. AB=4, AC=6, BC=20. Find BD.
HardCommon mistake
Mixing up which side of the ratio BD and DC belong to — BD (closer to B) corresponds to AB, and DC (closer to C) corresponds to AC; swapping them reverses the ratio.
Quick Review
- BD/DC = AB/AC, when AD bisects ∠A in △ABC.
- Proof idea: draw CE ∥ AD through C to create an isosceles triangle, then apply the basic proportionality theorem.
- Match BD to AB and DC to AC by which vertex (B or C) each segment is nearest to.