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+C1=0 and Px+Qy+C2=0, the equation of the angle bisector is given by:
    A(Px+Qy+C2)P(Ax+By+C1)A2+B2=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. 3x4y+5=0
  2. 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+C1=0 and Px+Qy+C2=0 is:

A(Px+Qy+C2)P(Ax+By+C1)A2+B2=0

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, C1=5, P=2, Q=5, and C2=7 into the formula:

3(2x+5y7)2(3x4y+5)32+(4)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 3x4y+5=0 and 2x+5y7=0.

Angle Bisectors of Lines in Slope-Intercept Form

Formula for Angle Bisector in Slope-Intercept Form:

  • For two lines given by y=mx+c1 and y=nx+c2, where m and n are slopes and c1 and c2 are y-intercepts:
    • The equation for the angle bisector between these lines is:
    y=m+n2x+c1+c22

Steps to Determine the Angle Bisector:

  1. Identify Slopes and Intercepts:

    • Determine the slopes (m and n) and y-intercepts (c1 and c2) of the given lines.
  2. Use the Angle Bisector Formula:

    • Substitute the slopes and intercepts into the formula for the angle bisector:
      y=m+n2x+c1+c22
  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, c1=3
    • For Line 2: n=3, c2=6
  2. Use the Angle Bisector Formula:

    • Substitute the slopes and intercepts into the formula for the angle bisector:
      y=232x+3+62
  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=m1x+c1 and y=m2x+c2, is given by:
    y=m1+m2m1×m2x+m1c2+m2c1m1+m2

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 m1+m2m1×m2.
  3. Intercept Calculation:

    • The y-intercept of the angle bisector line is m1c2+m2c1m1+m2.

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: m1=2 and c1=3
    • For Line 2: m2=3 and c2=6
  2. Apply the Angle Bisector Formula:

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