Angle Bisector Theorem

The Angle Bisector Theorem is a fundamental concept in geometry that is frequently applied in various mathematical problems, especially in triangles. Whether you're a student preparing for exams or a math enthusiast exploring the world of geometry, understanding the Angle Bisector Theorem is essential. In this blog, we will explore the theorem's definition, its proof, examples, and problems, and provide a downloadable PDF to help you master this topic.

1.0What is the Angle Bisector Theorem?

The Angle Bisector Theorem states that the angle bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the adjacent sides of the triangle. This theorem is a powerful tool when solving problems related to triangles, and it is widely used in geometry.

In simple terms:

If you have a triangle ABC with an angle bisector of $\angle A$ meeting the opposite side BC at point D, then the following relation holds:

This means that the ratio of the lengths of the two segments created on side BC is equal to the ratio of the lengths of the other two sides of the triangle, AB and AC.

2.0Triangle Angle Bisector Theorem Explained

Consider a triangle △ABC with angle $\angle A$. Let the angle bisector of $\angle A$ intersect the opposite side BC at point D. According to the Angle Bisector Theorem, the lengths of the segments BD and DC on side BC will satisfy the following proportional relationship:

$\frac{AB}{CD}=\frac{AC}{CD}$

Key Points to Remember:

• Angle Bisector: The angle bisector divides the angle into two equal parts.
• Proportional Segments: The segments created on the opposite side are proportional to the other two sides of the triangle.
• Applications: This theorem is particularly useful in solving geometry problems related to triangles, such as finding unknown side lengths or areas.

3.0Angle Bisector Theorem Proof

The Angle Bisector Theorem is often proven using similarity of triangles. Here's an outline of the proof:

1. Consider a triangle △ABC where the angle bisector of $\angle A$ meets the side BC at point D.
2. Draw a line parallel to side AB from point C, intersecting side AD at point E, so that .

Use the properties of similar triangles △ABD and △DCE.

In △ABD and △DCE  

∠BAD = ∠CED {Alternate Interior Angle}

∠BDA = ∠CED {Vertically Opposite Angle}

$\therefore$ By AA(Angle-Angle) criterion,

△ABD ~ △DCE  

From the similarity of triangles △ABD∼△DCE, we get the proportion:

$\frac{AB}{BD}=\frac{CE}{CD}$ … (A)

As we know AD is angle bisector of angle A, 

So, ∠BAD = ∠ CAD …(i)

And ∠BAD = ∠ CED {Alternate Interior Angle} …(ii)

From (i) and (ii)

∠CAD = ∠ CED

$\therefore$ △ACE is isosceles Triangle

Therefore, AC = CE

Substituting in (A)

$\frac{AB}{BD}=\frac{AC}{CD}$

This proof shows that the angle bisector divides the opposite side in a way that is directly proportional to the other two sides of the triangle.

4.0Angle Bisector Theorem Examples

Example 1: In triangle △ABC, the angle bisector of ∠A intersects side BC at point D. If AB = 6 cm, AC = 9 cm, and BD = 4 cm, what is the length of DC?

Solution: 

Using the Angle Bisector Theorem:

$\frac{AB}{BD}=\frac{AC}{CD}$

Substitute the known values:

$\frac{6}{4}=\frac{9}{DC}$

Solving for DC:

$\frac{3}{2}=\frac{9}{DC}\Rightarrow DC=\frac{9\times 2}{3}=6cm$

Thus, the length of DC is 6 cm.

Example 2: In a triangle △ABC, the angle bisector of ∠A divides side BC into segments BD = 8 cm and DC = 12 cm. If the length of AB is 10 cm, what is the length of AC?

Solution: 

Using the Angle Bisector Theorem:

$\frac{AB}{BD}=\frac{AC}{CD}$

Substitute the known values:

$\frac{10}{8}=\frac{AC}{12}$ 

Solving for AC:

$\frac{5}{4}=\frac{AC}{12}\Rightarrow =\frac{12\times 5}{4}=15$

Therefore, the length of AC is 15 cm.

5.0Angle Bisector Theorem Problems

To get more practice, try solving the following Angle Bisector Theorem problems:

1. In triangle △ABC, $\angle A=60^0$, AB = 10 cm, and AC = 15 cm. The angle bisector of ∠A divides BC into segments BD = 5 cm. Find the length of DC.
2. In triangle △XYZ, XY = 12 cm, XZ = 18 cm, and the angle bisector of ∠X intersects side YZ at point P. If YP = 8 cm, find the length of PZ.