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Angle Bisector Theorem: find BD and DC step by step

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Question

In triangle ABC, AB = 6 cm, AC = 4 cm and BC = 5 cm. AD bisects ∠A and meets BC at D. Find BD and DC.

Angle Bisector Trap: It doesn't split the base 50/50Watch on YouTube

We need to determine BD and DC from the three given side lengths and the fact that AD bisects angle A. We will first form the correct ratio, convert that ratio into actual lengths and then verify both the total length and the theorem before stating the answer.

ABCDAB = 6AC = 4BDDCBC = 5
AD halves angle A, but it does not usually halve BC. The two base pieces must follow the adjacent-side ratio, which is derived step by step below.

Step 1: write the Angle Bisector Theorem

The Angle Bisector Theorem states that an angle bisector in a triangle divides the opposite side into segments proportional to the adjacent sides:

BDDC=ABAC[By Angle Bisector Theorem: AD bisects ∠A]\frac{BD}{DC} = \frac{AB}{AC} \quad [\text{By Angle Bisector Theorem: } AD \text{ bisects } \angle A]
Substitute given lengths: AB=6 cm,  AC=4 cm\text{Substitute given lengths: } AB = 6\text{ cm},\; AC = 4\text{ cm}
BDDC=64\frac{BD}{DC} = \frac{6}{4}
Simplify fraction: BDDC=6÷24÷2=32\text{Simplify fraction: } \frac{BD}{DC} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2}
  ⟹  BD:DC=3:2\implies BD : DC = 3 : 2

Step 2: convert the ratio into actual lengths

Let each ratio part represent kk cm. Therefore, BD=3kBD = 3k and DC=2kDC = 2k.

Let BD=3k and DC=2k\text{Let } BD = 3k \text{ and } DC = 2k
Total length of base: BC=BD+DC\text{Total length of base: } BC = BD + DC
5=3k+2k5 = 3k + 2k
5=5k5 = 5k
55=5k5[Divide both sides by 5]\frac{5}{5} = \frac{5k}{5} \quad [\text{Divide both sides by } 5]
k=1k = 1
Calculate segment BD:BD=3k=3(1)=3 cm\text{Calculate segment } BD: BD = 3k = 3(1) = 3\text{ cm}
Calculate segment DC:DC=2k=2(1)=2 cm\text{Calculate segment } DC: DC = 2k = 2(1) = 2\text{ cm}

Step 3: verify the result

A valid solution must satisfy both the total segment addition and the theorem ratio:

Sum of parts: BD+DC=3 cm+2 cm=5 cm\text{Sum of parts: } BD + DC = 3\text{ cm} + 2\text{ cm} = 5\text{ cm}
Total base matches: 5 cm=BC(Sum verified ✓)\text{Total base matches: } 5\text{ cm} = BC \quad (\text{Sum verified } \checkmark)
Base ratio: BDDC=32=1.5\text{Base ratio: } \frac{BD}{DC} = \frac{3}{2} = 1.5
Adjacent side ratio: ABAC=64=32=1.5\text{Adjacent side ratio: } \frac{AB}{AC} = \frac{6}{4} = \frac{3}{2} = 1.5
BDDC=ABAC=32(Theorem verified ✓)\frac{BD}{DC} = \frac{AB}{AC} = \frac{3}{2} \quad (\text{Theorem verified } \checkmark)

Four checks before writing the final answer

  • Pair the correct sides. BD goes with AB; DC goes with AC.
  • Compare sizes. Because AB is longer than AC, BD must be longer than DC.
  • Add the pieces. The two answers must total BC.
  • Do not assume a half-and-half split. Equal base pieces occur only when the adjacent sides are equal.

Final step: state the verified lengths

BD=3 cm,DC=2 cm\boxed{BD = 3\text{ cm},\quad DC = 2\text{ cm}}

Young Math Brains helps you pair each base segment with the correct adjacent side, form the ratio and verify the final lengths.

Practise angle-bisector questions

Your turn

Without calculating, which base segment must be longer: BD or DC, and which given side tells you that?

Tell us in the comments on YouTube or Instagram, or email connect@youngmathbrains.com. We read every one.

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