Angle Bisector Formula

Before going to learn the angle bisector formula, let us recall what is an angular bisector. An angle bisector is a ray that divides an angle into exactly two equal halves. In a triangle, an angle bisector also divides the opposite side in the ratio of the sides containing the angle. Let us learn about the angle bisector formula with a few solved examples in the end.

What Is the Angle Bisector Formula?

Triangle angle bisector theorem states that "In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle". i.e., the angle bisector formula is:

\(\dfrac{\text{BD}}{\text{DC}} = \dfrac{\text{AB}}{\text{AC}}\) 

The easy method to remember the angle bisector theorem of a triangle is

Let us see the applications of the angle bisector formula in the section below.

Examples Using Angle Bisector Formula

Example 1: A triangle ABC has to be divided into two parts by a line AD such that the angles BAD and CAD are equal. If AB = 5 units, AC = 4 units, CD = 2 units, find the length of BD.

Solution:

It is given that the angles BAD and CAD are equal.

Hence AD is an angle bisector.

So, by angle bisector formula,

\[\dfrac{\text{BD}}{\text{DC}} = \dfrac{\text{AB}}{\text{AC}} \\[0.2cm]
\dfrac{\text{BD}}{\text{2}} = \dfrac{\text{5}}{\text{4}} \\[0.2cm]
\text{BD} = \dfrac{\text{5}}{\text{2}} \]

Answer: \(\text{BD} = \dfrac{\text{5}}{\text{2}}\)

Example 2: A \({\triangle}\text{ABC}\) right angled at B has the side lengths AB and BC as 3 and 4 units respectively if the angle bisector of angle A meets the side BC at D such that the length BD is equal to \(x\). Find the value of \(x\).

Solution:

Given that \({\triangle}\text{ABC}\) right angled at B.

AB = 3
BC = 4

Let's find AC using the Pythagoras theorem.

\[\begin{align} {AC}^2 &= {BC}^2 + {AB}^2 \\[0.2cm]
{AC}^2 &= 4^2 + 3^2 \\[0.2cm]
{AC}^2 &= 16 + 9 \\[0.2cm]
{AC}^2 &= 25 \\[0.2cm]
AC &= 5 \end{align}\]

It is also given that AD is the angle bisector of angle A.

By, angle bisector formula:

\[\begin{align} \frac{\text{CD}}{\text{DB}} &= \frac{\text{AC}}{\text{AB}} \\[0.2cm]
\frac{4 - x}{x} &= \frac{5}{3} \\[0.2cm]
3(4 - x) &= 5x \\[0.2cm]
12 - 3x &= 5x \\[0.2cm]
12 &= 8x \\[0.2cm]
3 &= 2x \\[0.2cm]
\frac{3}{2} &= x \end{align}\]

\(\therefore x = \dfrac{3}{2}\)                                                                                                        

Answer: \(x = \dfrac{3}{2}\)