Triangle Angle Bisector Problem: Find the Segments

A classic geometry problem, worked out step by step by Dr. E — with the full reasoning behind every move, a video walkthrough, and a free practice worksheet.

📐 Geometry🎬 Video solution📄 Free worksheet
The Problem
A B C D 8 6 7
Figure not drawn to scale

In triangle ABC, the bisector of ∠A meets side BC at point D. Given AB = 8, AC = 6, and BC = 7, find the lengths of BD and DC.

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Video solution coming soon. Dr. E walks through this exact problem on the whiteboard, step by step.

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Step-by-Step Solution
1
Recall the Angle Bisector Theorem.
When the bisector of an angle of a triangle meets the opposite side, it divides that side into two segments proportional to the two sides forming the angle. The bisector from A meets BC at D, so BD / DC = AB / AC = 8 / 6 = 4 / 3.
2
Split BC using that ratio.
The two segments are in the ratio 4 : 3, so picture BC cut into 4 + 3 = 7 equal parts. Since BC = 7, each part is exactly 1 unit long.
3
Read off each segment.
BD = (4/7) · 7 = 4 and DC = (3/7) · 7 = 3.
Check: 4 + 3 = 7 = BC. ✓
Answer BD = 4, DC = 3 The angle bisector from a vertex divides the opposite side in the same ratio as the two adjacent sides — here 8 : 6, which reduces to 4 : 3.

💡 Why this works — the picture to remember

The whole problem is one rule: the bisector copies the side ratio onto the base. The longer side (AB = 8) "pulls" the point D toward it, so the bigger segment (BD) sits next to the bigger side. Once you see BD : DC = 8 : 6, the rest is just sharing 7 units in that ratio.

Great extension question: with an angle or the Law of Cosines you can find the actual length of the bisector AD itself — a natural next step for stronger students.

📄 Practice this skill — free worksheet Angle bisectors & perpendicular bisectors PDF — more problems just like this one. 📐 All geometry video lessons 7 units from foundations to circles — free, in English and Filipino. 🎬 Numberbender on YouTube 700+ free math lessons by Dr. E. Subscribe for new solved problems.
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