The Angle Bisector Theorem

An angle bisector splits the opposite side of a triangle in the ratio of the two adjacent sides. Learn the theorem and how to use it to find missing lengths.

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An angle bisector of a triangle splits one of its angles into two equal halves and meets the opposite side. It divides that opposite side in a very predictable way.

Theorem The Angle Bisector Theorem

If AD bisects ∠BAC, then D divides the opposite side so that BD : DC equals AB : AC.

A B C D

Example Finding a length

In triangle ABC, AB = 6, AC = 14, and BC = 18. AD bisects ∠BAC. Find BD.

Since BD + DC = 18:
⟹ BD = 5.4 and DC = 12.6.
Example Finding a side

In triangle PQR, PQ = 9. The bisector from P meets QR at S, with QS = 6 and SR = 9. Find PR.

⟹ PR = 13.5.
Summary
  1. An angle bisector splits the opposite side in the ratio of the two adjacent sides: BD : DC = AB : AC.
  2. Always remember that BD + DC = BC — the two parts make up the whole side.
  3. The three angle bisectors of a triangle meet at one point, the incenter (centre of the inscribed circle).
  4. Common slip: don't mix up the two adjacent sides with the opposite side.
283. Relations and Functions
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