3.2.7
Equation of a Perpendicular Bisector
Perpendicular Bisector of a Chord
Perpendicular Bisector of a Chord
We can find the equation of the perpendicular bisector of a chord using the properties of straight lines graphs.
![Illustrative background for What is the perpendicular bisector of a chord?](https://image-v2.cdn.app.senecalearning.com/2020-05/eb5eb103-3863-474b-aa51-64d4a1c0066d/Screenshot%202020-05-28%20at%2012.24.16-min,h_400,q_80,w_640.png)
![Illustrative background for What is the perpendicular bisector of a chord? ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/eb5eb103-3863-474b-aa51-64d4a1c0066d/Screenshot%202020-05-28%20at%2012.24.16-min,h_400,q_80,w_640.png)
What is the perpendicular bisector of a chord?
What is the perpendicular bisector of a chord?
- The perpendicular bisector of a chord is the perpendicular line that passes through the midpoint of the chord.
- The perpendicular bisector of a chord always passes through the centre of the circle.
![Illustrative background for What is the gradient of the perpendicular bisector?](https://image-v2.cdn.app.senecalearning.com/2020-05/ddef0a34-ab26-4e44-bea4-7ec84f459660/Screenshot%202020-05-28%20at%2012.24.23-min,h_400,q_80,w_640.png)
![Illustrative background for What is the gradient of the perpendicular bisector? ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/ddef0a34-ab26-4e44-bea4-7ec84f459660/Screenshot%202020-05-28%20at%2012.24.23-min,h_400,q_80,w_640.png)
What is the gradient of the perpendicular bisector?
What is the gradient of the perpendicular bisector?
- We first need to find the gradient of the chord using the equation:
- Where and are the points where the chord meets the circumference.
- The bisector is perpendicular to the chord, so its gradient satisfies:
![Illustrative background for What is the equation of the perpendicular bisector?](https://image-v2.cdn.app.senecalearning.com/2020-05/ca526027-782b-4bdf-97ec-6eca2f27adb3/Screenshot%202020-05-28%20at%2012.24.48-min,h_400,q_80,w_640.png)
![Illustrative background for What is the equation of the perpendicular bisector? ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/ca526027-782b-4bdf-97ec-6eca2f27adb3/Screenshot%202020-05-28%20at%2012.24.48-min,h_400,q_80,w_640.png)
What is the equation of the perpendicular bisector?
What is the equation of the perpendicular bisector?
- We can then find the equation of the bisector by working out the midpoint of the chord:
- Midpoint =
- Finally, we substitute this into the general equation of a straight line with gradient to find the
-intercept
:
![Illustrative background for Example](https://image-v2.cdn.app.senecalearning.com/2020-05/ca614d60-7483-4578-87a0-cafbcfef35ab/Screenshot%202020-05-28%20at%2012.25.07-min,h_400,q_80,w_640.png)
![Illustrative background for Example ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/ca614d60-7483-4578-87a0-cafbcfef35ab/Screenshot%202020-05-28%20at%2012.25.07-min,h_400,q_80,w_640.png)
Example
Example
- The points and lie on the circumference of a circle.
- What is the equation of the perpendicular bisector of the chord AB?
![Illustrative background for Find the gradient of chord](https://image-v2.cdn.app.senecalearning.com/2020-05/203f215a-8b0f-4a1c-a180-f0e4110a5a9b/Screenshot%202020-05-28%20at%2013.45.05-min,h_400,q_80,w_640.png)
![Illustrative background for Find the gradient of chord ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/203f215a-8b0f-4a1c-a180-f0e4110a5a9b/Screenshot%202020-05-28%20at%2013.45.05-min,h_400,q_80,w_640.png)
Find the gradient of chord
Find the gradient of chord
- The gradient of the chord is given by:
![Illustrative background for Find the gradient of bisector](https://image-v2.cdn.app.senecalearning.com/2020-05/e144ea0b-65df-48e0-9ec2-da05b744f381/Screenshot%202020-05-28%20at%2013.47.34-min,h_400,q_80,w_640.png)
![Illustrative background for Find the gradient of bisector ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/e144ea0b-65df-48e0-9ec2-da05b744f381/Screenshot%202020-05-28%20at%2013.47.34-min,h_400,q_80,w_640.png)
Find the gradient of bisector
Find the gradient of bisector
- The gradient of the perpendicular bisector satisfies the equation:
![Illustrative background for Find the midpoint of chord](https://image-v2.cdn.app.senecalearning.com/2020-05/0fec9910-0117-4fc9-83ee-8f59f30a9ff8/Screenshot%202020-05-28%20at%2012.26.26-min,h_400,q_80,w_640.png)
![Illustrative background for Find the midpoint of chord ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/0fec9910-0117-4fc9-83ee-8f59f30a9ff8/Screenshot%202020-05-28%20at%2012.26.26-min,h_400,q_80,w_640.png)
Find the midpoint of chord
Find the midpoint of chord
- The midpoint of the chord AB is equal to:
- midpoint =
![Illustrative background for Find the y-intercept of the perpendicular bisector](https://image-v2.cdn.app.senecalearning.com/2020-05/b84e1d99-8060-4aed-be2f-42be7b5ace4f/Screenshot%202020-05-28%20at%2013.49.47-min,h_400,q_80,w_640.png)
![Illustrative background for Find the y-intercept of the perpendicular bisector ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/b84e1d99-8060-4aed-be2f-42be7b5ace4f/Screenshot%202020-05-28%20at%2013.49.47-min,h_400,q_80,w_640.png)
Find the y-intercept of the perpendicular bisector
Find the y-intercept of the perpendicular bisector
- Substitute and the midpoint into the general equation of a straight line:
- Rearranging for , we get:
![Illustrative background for Final answer](https://image-v2.cdn.app.senecalearning.com/2020-05/3608cb98-3827-425f-9237-d3a23b9ac5b0/Screenshot%202020-05-28%20at%2012.27.25-min,h_400,q_80,w_640.png)
![Illustrative background for Final answer ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/3608cb98-3827-425f-9237-d3a23b9ac5b0/Screenshot%202020-05-28%20at%2012.27.25-min,h_400,q_80,w_640.png)
Final answer
Final answer
- So the equation of the perpendicular bisector of the chord AB is:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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