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:
$$\frac{A(Px+Qy+{C}_{2})P(Ax+By+{C}_{1})}{\sqrt{{A}^{2}+{B}^{2}}}=0$$
Characteristics of Angle Bisectors:
 Equal Division:
 Angle bisectors divide the angle created by the intersection of two lines into two equal parts.
 Equal Distance:
 They maintain an equal distance from both intersecting lines.
 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:
 $3x4y+5=0$
 $2x+5y7=0$
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:
Step 2: Calculate Bisector Equation
Using the given lines $3x4y+5=0$ and $2x+5y7=0$, 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:
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 $3x4y+5=0$ and $2x+5y7=0$.
Angle Bisectors of Lines in SlopeIntercept Form
Formula for Angle Bisector in SlopeIntercept 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 yintercepts:
 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:

Identify Slopes and Intercepts:
 Determine the slopes ($m$ and $n$) and yintercepts (${c}_{1}$ and ${c}_{2}$) of the given lines.

Use the Angle Bisector Formula:
 Substitute the slopes and intercepts into the formula for the angle bisector:
$$y=\frac{m+n}{2}\cdot x+\frac{{c}_{1}+{c}_{2}}{2}$$
 Substitute the slopes and intercepts into the formula for the angle bisector:

Simplify the Equation:
 Simplify the equation to obtain the equation representing the angle bisector between the given lines.
Example:
Given lines:
 $y=2x+3$ (Line 1)
 $y=3x+6$(Line 2)
Determining the Angle Bisector:

Identify Slopes and Intercepts:
 For Line 1: $m=2$, ${c}_{1}=3$
 For Line 2: $n=3$, ${c}_{2}=6$

Use the Angle Bisector Formula:
 Substitute the slopes and intercepts into the formula for the angle bisector:
$$y=\frac{23}{2}\cdot x+\frac{3+6}{2}$$
 Substitute the slopes and intercepts into the formula for the angle bisector:

Simplify the Equation:
 Calculate to simplify the equation to represent the angle bisector between the given lines.
Angle Bisectors of Lines in SlopeIntercept Form
Introduction:
 Angle bisectors of lines are lines that divide the angle formed by two intersecting lines into two equal angles.
 Representing lines in slopeintercept form ($y=mx+c$) simplifies determining angle bisectors between lines.
Formula for Angle Bisectors in SlopeIntercept Form:
 The equation of the angle bisector between lines in slopeintercept form, $y={m}_{1}x+{c}_{1}$ and $y={m}_{2}x+{c}_{2}$, is given by:
$$y=\frac{{m}_{1}+{m}_{2}}{{m}_{1}\times {m}_{2}}\cdot x+\frac{{m}_{1}{c}_{2}+{m}_{2}{c}_{1}}{{m}_{1}+{m}_{2}}$$
Characteristics of Angle Bisectors in SlopeIntercept Form:

Equal Division:
 Bisects the angle formed by the intersection of the two lines into two equal angles.

Slope Calculation:
 The slope of the angle bisector line is given by $\frac{{m}_{1}+{m}_{2}}{{m}_{1}\times {m}_{2}}$.

Intercept Calculation:
 The yintercept of the angle bisector line is $\frac{{m}_{1}{c}_{2}+{m}_{2}{c}_{1}}{{m}_{1}+{m}_{2}}$.
Example of Angle Bisectors in SlopeIntercept Form:
Given lines in slopeintercept form:
 $y=2x+3$ (Line 1)
 $y=3x+6$ (Line 2)
Determining the Angle Bisector:

Calculate Slopes and Intercepts:
 For Line 1: ${m}_{1}=2$ and ${c}_{1}=3$
 For Line 2: ${m}_{2}=3$ and ${c}_{2}=6$

Apply the Angle Bisector Formula:
 Use the formula to calculate the equation representing the angle bisector between the given lines.