Angle Bisectors of Two Lines

This math recipe will help you find the equations of the angle bisectors of two lines in the XY plane.

Easy

Approx. 2 min

Equations of the two lines: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

The equations of the angle bisectors are given by:

(a₁x + b₁y + c₁)/√(a₁² + b₁²) = ± (a₂x + b₂y + c₂)/√(a₂² + b₂²)

Example 1 Find the equation of angle bisectors for the pair of lines 3x + 4y – 7 = 0 and 4x + 3y – 7 = 0.

Solution To find the equations, we’ll apply the above formula directly. The required equations are:

(3x + 4y – 7)/√(3² + 4²) = ± (4x + 3y – 7)/√(4² + 3²)

On rearranging the terms, we’ll get the following two equations:

x – y = 0

x + y = 2

Here’s a figure that shows the two lines and their bisectors.

Example 2 Find the equation of angle bisectors for the pair of lines 5x + 12y + 7 = 0 and 12x – 5y – 17 = 0

Solution To find the equations, we’ll apply the above formula directly. The required equations are:

(5x + 12y + 7)/√(5² + 12²) = ± (12x – 5y – 17)/√(12² + (-5)²)

On rearranging the terms, we’ll get the following two equations:

17x + 7y – 10 = 0

7x – 17y – 24 = 0

Here’s a figure that shows the two lines and their bisectors.

That’s it for this recipe. Hope you found it helpful.

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