3.2.9
Circumcircle of a Right-angled Triangle
Circumcircle of a Right-Angled Triangle
Circumcircle of a Right-Angled Triangle
The circumcircle of a right-angled triangle is easily found by looking at the midpoint of its hypotenuse.
Triangle in a semi-circle
Triangle in a semi-circle
- There is a circle theorem that states that the angle of a triangle at the circumference of a semi-circle is always 90°.
- This means that the hypotenuse of a right-angled triangle is the diameter of its circumcircle.
- We can use this to find the equation of the circumcircle of the triangle.
How do you find the hypotenuse of the right-angled triangle?
How do you find the hypotenuse of the right-angled triangle?
- To identify which two vertices of the triangle form the hypotenuse, we first need to find the square of the length of each side.
- We can identify the hypotenuse using Pythagoras' theorem:
- Opposite side2 + adjacent side2 = hypotenuse2
How do you find the equation of the circumcircle?
How do you find the equation of the circumcircle?
- To write the general equation of the circumcircle, we need to find the centre and the radius of the circle.
- The centre is the midpoint of the hypotenuse.
- The radius is half the length of the hypotenuse.
Example
Example
- The points A(3,6), B(3,4) and C(4,5) lie on the circumference of a circle.
- Show that ABC is a right-angled triangle.
- Find the equation of the circle.
Use Pythagoras' theorem
Use Pythagoras' theorem
- All right-angled triangles satisfy Pythagoras' theorem.
- We can see that as 2 + 2 = 4.
- This means that is a right-angled triangle with as its hypotenuse.
Use circle theorem
Use circle theorem
- is a right-angled triangle, so is the diameter of its circumcircle.
Find the centre
Find the centre
- The centre of the circumcircle is the midpoint of the diameter .
- Midpoint =
Find the radius
Find the radius
- As calculated earlier, = 4. This means:
- Diameter = 2
- Radius = 1
Write the equation of the circle
Write the equation of the circle
- We can now write the equation of the circumcircle in its general form:
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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