What is the Equation of Angle Bisector? | Simple Explanation for Class 10

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What is the Equation of the Angle Bisector of Two Lines?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of the angle bisector of two lines represents a third line that divides the angle formed by the first two lines into two equal halves. Imagine cutting a slice of pizza exactly in half; the cut line is the angle bisector. There are always two angle bisectors for any two intersecting lines.

Simple Example

Quick Example

Think about two roads meeting at a junction, forming an angle. If you want to build a new road that perfectly splits the angle between these two roads, the path of that new road would be an angle bisector. It ensures that any point on the new road is equally far from both original roads.

Worked Example

Step-by-Step

Let's find the equation of the angle bisectors for the lines: L1: 3x + 4y - 5 = 0 and L2: 4x - 3y + 2 = 0.

Step 1: Write down the general formula for the angle bisector of two lines A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0. It is (A1x + B1y + C1) / sqrt(A1^2 + B1^2) = ± (A2x + B2y + C2) / sqrt(A2^2 + B2^2).
---Step 2: Identify the coefficients for L1. A1 = 3, B1 = 4, C1 = -5. Calculate sqrt(A1^2 + B1^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
---Step 3: Identify the coefficients for L2. A2 = 4, B2 = -3, C2 = 2. Calculate sqrt(A2^2 + B2^2) = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = sqrt(25) = 5.
---Step 4: Substitute these values into the formula for the '+' case.
(3x + 4y - 5) / 5 = (4x - 3y + 2) / 5
---Step 5: Simplify the equation by multiplying both sides by 5.
3x + 4y - 5 = 4x - 3y + 2
---Step 6: Rearrange the terms to get the first bisector equation.
4x - 3x - 3y - 4y + 2 + 5 = 0
x - 7y + 7 = 0. This is the equation of the first angle bisector.
---Step 7: Now substitute for the '-' case.
(3x + 4y - 5) / 5 = - (4x - 3y + 2) / 5
---Step 8: Simplify and rearrange.
3x + 4y - 5 = -4x + 3y - 2
3x + 4x + 4y - 3y - 5 + 2 = 0
7x + y - 3 = 0. This is the equation of the second angle bisector.

Answer: The equations of the angle bisectors are x - 7y + 7 = 0 and 7x + y - 3 = 0.

Why It Matters

Understanding angle bisectors is crucial in fields like robotics and computer graphics to precisely control movement and create symmetrical designs. Engineers use this concept when designing machine parts or planning the paths of autonomous vehicles, ensuring they maintain equal distance from boundaries. Architects might use it for symmetrical building layouts.

Common Mistakes

MISTAKE: Forgetting the '±' sign, which leads to finding only one bisector instead of two. | CORRECTION: Always remember that two intersecting lines have two angle bisectors (one acute, one obtuse), so you must use both the '+' and '-' signs in the formula.

MISTAKE: Incorrectly calculating the square root of the sum of squares in the denominator, especially with negative coefficients. For example, using -3^2 as -9. | CORRECTION: Remember that squaring a negative number always results in a positive number (e.g., (-3)^2 = 9). Be careful with arithmetic.

MISTAKE: Not bringing all terms to one side to get the standard form Ax + By + C = 0 for each bisector. | CORRECTION: After simplifying the equation, always move all terms to one side of the equality sign to express the final answer in the standard linear equation form.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the angle bisectors for the lines x + y - 1 = 0 and x - y + 2 = 0. | ANSWER: x + 2y - 3 = 0 and 2x + y + 1 = 0

QUESTION: Determine the equations of the bisectors for the lines 5x + 12y - 10 = 0 and 12x - 5y + 15 = 0. | ANSWER: 17x + 7y + 5 = 0 and 7x - 17y + 25 = 0

QUESTION: Two lines pass through the origin. One line is y = 2x and the other is y = -x. Find the equations of their angle bisectors. (Hint: Convert to standard form first). | ANSWER: 3x - y = 0 and x + 3y = 0

MCQ

Quick Quiz

Which of the following is the general formula for the angle bisector of two lines A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0?

(A1x + B1y + C1) / (A1 + B1) = ± (A2x + B2y + C2) / (A2 + B2)

(A1x + B1y + C1) / sqrt(A1^2 + B1^2) = ± (A2x + B2y + C2) / sqrt(A2^2 + B2^2)

A1x + B1y + C1 = A2x + B2y + C2

(A1x + B1y + C1)^2 = (A2x + B2y + C2)^2

The Correct Answer Is:

Option B is correct because the distance of a point (x,y) from a line Ax + By + C = 0 is given by |Ax + By + C| / sqrt(A^2 + B^2). An angle bisector is the locus of points equidistant from the two lines, hence the formula.

Real World Connection

In the Real World

In city planning, when designing new roads or railway lines, engineers often need to find a path that is equally distant from two existing paths or boundaries. For example, if two main roads meet, a new service road or a green belt might be planned along the angle bisector to maintain symmetry and equal access for residents, much like how Google Maps might suggest the most 'balanced' route between two points.

Key Vocabulary

Key Terms

Angle Bisector: A line that divides an angle into two equal parts. | Locus: The set of all points satisfying a given condition. | Standard Form of a Line: An equation of a line written as Ax + By + C = 0. | Coefficient: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

What's Next

What to Learn Next

Now that you understand angle bisectors, you can explore concepts like the incenter of a triangle, which is the intersection point of the angle bisectors of a triangle. This builds on your knowledge and helps you solve more complex geometry problems!