# Angle Bisectors of Lines

### Angle Bisectors of Lines

#### Definition:

• Angle bisectors of lines are lines that divide the angle formed by two intersecting lines into two equal angles.

#### Method to Determine Angle Bisectors:

• Given two lines with equations $Ax+By+{C}_{1}=0$ and $Px+Qy+{C}_{2}=0$, the equation of the angle bisector is given by:
$\frac{A\left(Px+Qy+{C}_{2}\right)-P\left(Ax+By+{C}_{1}\right)}{\sqrt{{A}^{2}+{B}^{2}}}=0$

#### Characteristics of Angle Bisectors:

1. Equal Division:
• Angle bisectors divide the angle created by the intersection of two lines into two equal parts.
2. Equal Distance:
• They maintain an equal distance from both intersecting lines.
3. Perpendicularity:
• Angle bisectors are perpendicular to the line joining the intersection point of the given lines to the point of intersection of the angle bisector.

### Example:

Given lines:

1. $3x-4y+5=0$
2. $2x+5y-7=0$

### Determining the Angle Bisector:

#### Step 1: Equation of the Angle Bisector

The equation for the angle bisector between two lines $Ax+By+{C}_{1}=0$ and $Px+Qy+{C}_{2}=0$ is:

$\frac{A\left(Px+Qy+{C}_{2}\right)-P\left(Ax+By+{C}_{1}\right)}{\sqrt{{A}^{2}+{B}^{2}}}=0$

#### Step 2: Calculate Bisector Equation

Using the given lines $3x-4y+5=0$ and $2x+5y-7=0$, apply the formula for the angle bisector:

Substitute $A=3$, $B=-4$, ${C}_{1}=5$, $P=2$, $Q=5$, and ${C}_{2}=-7$ into the formula:

$\frac{3\left(2x+5y-7\right)-2\left(3x-4y+5\right)}{\sqrt{{3}^{2}+\left(-4{\right)}^{2}}}=0$

#### Step 3: Simplify the Equation

Solve the equation to determine the equation representing the angle bisector between the given lines.

#### Conclusion:

Upon solving the equation, the resulting equation represents the line that bisects the angle formed by the intersection of the given lines $3x-4y+5=0$ and $2x+5y-7=0$.

### Angle Bisectors of Lines in Slope-Intercept Form

#### Formula for Angle Bisector in Slope-Intercept Form:

• For two lines given by $y=mx+{c}_{1}$ and $y=nx+{c}_{2}$, where $m$ and $n$ are slopes and ${c}_{1}$ and ${c}_{2}$ are y-intercepts:
• The equation for the angle bisector between these lines is:
$y=\frac{m+n}{2}\cdot x+\frac{{c}_{1}+{c}_{2}}{2}$

#### Steps to Determine the Angle Bisector:

1. Identify Slopes and Intercepts:

• Determine the slopes ($m$ and $n$) and y-intercepts (${c}_{1}$ and ${c}_{2}$) of the given lines.
2. Use the Angle Bisector Formula:

• Substitute the slopes and intercepts into the formula for the angle bisector:
$y=\frac{m+n}{2}\cdot x+\frac{{c}_{1}+{c}_{2}}{2}$
3. Simplify the Equation:

• Simplify the equation to obtain the equation representing the angle bisector between the given lines.

#### Example:

Given lines:

1. $y=2x+3$ (Line 1)
2. $y=-3x+6$ (Line 2)

#### Determining the Angle Bisector:

1. Identify Slopes and Intercepts:

• For Line 1: $m=2$, ${c}_{1}=3$
• For Line 2: $n=-3$, ${c}_{2}=6$
2. Use the Angle Bisector Formula:

• Substitute the slopes and intercepts into the formula for the angle bisector:
$y=\frac{2-3}{2}\cdot x+\frac{3+6}{2}$
3. Simplify the Equation:

• Calculate to simplify the equation to represent the angle bisector between the given lines.

### Angle Bisectors of Lines in Slope-Intercept Form

#### Introduction:

• Angle bisectors of lines are lines that divide the angle formed by two intersecting lines into two equal angles.
• Representing lines in slope-intercept form ($y=mx+c$) simplifies determining angle bisectors between lines.

#### Formula for Angle Bisectors in Slope-Intercept Form:

• The equation of the angle bisector between lines in slope-intercept form, $y={m}_{1}x+{c}_{1}$ and $y={m}_{2}x+{c}_{2}$, is given by:
$y=\frac{{m}_{1}+{m}_{2}}{{m}_{1}×{m}_{2}}\cdot x+\frac{{m}_{1}{c}_{2}+{m}_{2}{c}_{1}}{{m}_{1}+{m}_{2}}$

#### Characteristics of Angle Bisectors in Slope-Intercept Form:

1. Equal Division:

• Bisects the angle formed by the intersection of the two lines into two equal angles.
2. Slope Calculation:

• The slope of the angle bisector line is given by $\frac{{m}_{1}+{m}_{2}}{{m}_{1}×{m}_{2}}$.
3. Intercept Calculation:

• The y-intercept of the angle bisector line is $\frac{{m}_{1}{c}_{2}+{m}_{2}{c}_{1}}{{m}_{1}+{m}_{2}}$.

#### Example of Angle Bisectors in Slope-Intercept Form:

Given lines in slope-intercept form:

1. $y=2x+3$ (Line 1)
2. $y=-3x+6$ (Line 2)

#### Determining the Angle Bisector:

1. Calculate Slopes and Intercepts:

• For Line 1: ${m}_{1}=2$ and ${c}_{1}=3$
• For Line 2: ${m}_{2}=-3$ and ${c}_{2}=6$
2. Apply the Angle Bisector Formula:

• Use the formula to calculate the equation representing the angle bisector between the given lines.