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The angle bisector theorem

الأسبوع 10 · منصفا الزاوية والأجزاء المتناسبة: المنصف الداخلي والخارجي وطول المنصف · 3 دروس
1

The bisector of an interior angle divides the opposite side in the ratio of the other two sides: BD/DC = AB/AC

Mathematics Sec 1 Term 1, Unit 3, Lesson 2
GoalUse the angle-bisector theorem to find the parts of the opposite side.
New words
angle bisector · the ray that cuts an angle into two equal anglesinterior bisector · the bisector of an angle inside the triangle, meeting the opposite side
Explanation

Theorem: the bisector of an interior angle of a triangle divides the opposite side into two parts whose ratio equals the ratio of the other two sides. If AD bisects ∠A with D on BC, then BD/DC = AB/AC. Sketch of the reason: draw through C a line parallel to AD meeting BA extended at E. Then ∠E = ∠BAD (corresponding) and ∠ACE = ∠DAC (alternate), and since the bisector makes ∠BAD = ∠DAC, △ACE is isosceles with AE = AC. The parallel line theorem in △BCE gives BD/DC = BA/AE = BA/AC. The ratio uses the two sides that meet at the bisected angle (AB and AC), never BC. Note that D is not the midpoint of BC unless AB = AC. To find the parts, call them in the given ratio and use BD + DC = BC.

ƒFormula
BD/DC = AB/AC
AD bisects ∠A: the ratio of the two parts of BC is the ratio of the two sides that meet at A
Worked examples
10612ABCD
AD bisects ∠A in △ABC: AB = 10, AC = 6 and BC = 12. BD/DC = 10/6 = 5/3. BD + DC = 12 is 8 parts, so one part is 1.5: BD = 5 × 1.5 = 7.5 cm and DC = 3 × 1.5 = 4.5 cm.
AD bisects ∠A with BD = 9, DC = 15 and the perimeter of △ABC is 72 cm. BC = 24, so AB + AC = 48. AB/AC = 9/15 = 3/5, so AB = 3k, AC = 5k and 8k = 48, k = 6. AB = 18 cm and AC = 30 cm.
△ABC has a right angle at B, AD bisects ∠A, BA : AC = 4 : 5 and BD = 8. BD/DC = 4/5, so DC = 10 and BC = 18. The sides are in the ratio 4 : 3 : 5 (BA : BC : AC), so BC = 3k = 18, k = 6: AB = 24 and AC = 30. Perimeter 24 + 18 + 30 = 72 cm.
Class activity

Fold a bisector. Cut a paper triangle with sides about 10, 6 and 12 cm. Fold it so that AB lies on AC to make the bisector of the angle at A. Measure BD and DC and compare BD : DC with AB : AC.

Practice
1
AD bisects ∠A: AB = 12, AC = 8 and BC = 10. Find BD and DC.
2
AD bisects ∠A: AB = 9, AC = 15 and BC = 16. Find BD and DC.
3
AD bisects ∠A: BD = 5, DC = 7 and AB = 10. Find AC.
4
AD bisects ∠A: BD = 6, DC = 4 and the perimeter of △ABC is 45. Find AB and AC.
5
Is D the midpoint of BC when AB ≠ AC? Explain.