Overview

Angle bisectors turn angles into side ratios. They are essential for area and segment-ratio problems.

Key Ideas

• If $AD$ bisects $\angle A$ in triangle $ABC$, then $\frac{BD}{DC} = \frac{AB}{AC}$.
• Triangles $ABD$ and $ACD$ share a height from $A$, so their areas are in the ratio $BD:DC$.
• The bisector from a vertex lies inside the triangle and hits the opposite side between the endpoints.

Core Skills

Apply the Angle Bisector Theorem

Set up $BD/DC = AB/AC$ and use $BD+DC=BC$ when a full side length is given.

Use Area Ratios

Because $ABD$ and $ACD$ share an altitude from $A$, their areas are proportional to $BD$ and $DC$. This is a fast shortcut when areas are known.

Check Interior vs Exterior

If the bisector is exterior, the ratio flips to involve signed segments. In Foundations, most problems use the interior bisector.

Worked Example

In triangle $ABC$, $AB=8$, $AC=6$, and $BC=7$. The bisector from $A$ meets $BC$ at $D$. Find $BD$.

$BD/DC = 8/6 = 4/3$ and $BD+DC=7$. So $BD=4$ and $DC=3$.

More Examples

Example 1: Quick Ratio

In triangle $ABC$, $AB=5$ and $AC=10$. The angle bisector from $A$ meets $BC$ at $D$. Find $BD:DC$.

$BD:DC = AB:AC = 1:2$.

Example 2: Area Ratio

If $[ABD]=12$ and $[ACD]=18$, find $BD:DC$.

The areas share height from $A$, so $BD:DC=12:18=2:3$.

Strategy Checklist

• Confirm the segment is an angle bisector before using the ratio.
• Use $BD+DC=BC$ to solve for segments when needed.
• Convert area information into side ratios quickly.

Common Pitfalls

• Applying the theorem to an exterior bisector without adjusting the ratio.
• Mixing side lengths and segment lengths.
• Forgetting that the ratio uses the adjacent sides to the bisected angle.

Practice Problems