3.2.8

Equation of a Circumcircle

Test yourself

Equation of a Circumcircle of a Triangle from Three Points

We can find the general equation of a circumcircle of a triangle by using the properties of chords.

Illustrative background for What is a circumcircle?Illustrative background for What is a circumcircle? ?? "content

What is a circumcircle?

  • The circumcircle of a triangle is the circumference of a circle upon which the three vertices of the triangle lie.
    • A group of three points can only have one circumcircle.
Illustrative background for What is a circumcircle?Illustrative background for What is a circumcircle? ?? "content

What is a circumcircle?

  • Each side of the triangle is a chord of the circle.
  • The perpendicular bisectors of the triangle sides intersect at the centre of the circle.
Illustrative background for How do you find the equation of a circumcircle?Illustrative background for How do you find the equation of a circumcircle? ?? "content

How do you find the equation of a circumcircle?

  • To write the general equation of the circumcircle, we need to find the centre and the radius of the circle.
Illustrative background for How do you find the equation of a circumcircle?Illustrative background for How do you find the equation of a circumcircle? ?? "content

How do you find the equation of a circumcircle?

  • The centre is found by working out the point of intersection between the perpendicular bisectors of two of the sides of the triangle.
  • The radius is found by working out the distance between the centre of the circle and one of the points of the triangle.
Illustrative background for ExampleIllustrative background for Example ?? "content

Example

  • The points A(-6,3), B(-3,2) and C(0,3) lie on the circumference of a circle.
    • What is the equation of the circle?
Illustrative background for Find the perpendicular bisector of ABIllustrative background for Find the perpendicular bisector of AB ?? "content

Find the perpendicular bisector of AB

  • The gradient of the chord AB is equal to:
    • 23(3)(6)=13 \frac{2-3}{(-3) - (-6)}=-\frac{1}{3}
  • The gradient of the perpendicular bisector of AB must be equal to 3.
  • The midpoint of AB has coordinates:
    • Midpoint = ((6)+(3)2,3+22)=(4.5,2.5)(\frac{(-6)+(-3)}{2},\frac{3+2}{2}) = (-4.5,2.5)
Illustrative background for Find the perpendicular bisector of ABIllustrative background for Find the perpendicular bisector of AB ?? "content

Find the perpendicular bisector of AB

  • The intercept of the perpendicular bisector, cc is then found from:
    • y=3x+cy = 3x + c
Illustrative background for Find the perpendicular bisector of ABIllustrative background for Find the perpendicular bisector of AB ?? "content

Find the perpendicular bisector of AB

  • Substituting in the midpoint coordinates, we get:
    • 2.5=(3×(4.5))+c2.5 = (3 \times (-4.5)) + c
    • 2.5=13.5+cc=162.5 = -13.5 +c \rightarrow c = 16
  • So the equation of the perpendicular bisector of AB is y=3x+16y = 3x +16
Illustrative background for Find the perpendicular bisector of ACIllustrative background for Find the perpendicular bisector of AC ?? "content

Find the perpendicular bisector of AC

  • The gradient of the chord AC is equal to:
    • 330(6)=0 \frac{3-3}{0 - (-6)}=0
  • This means that AC is horizontal.
  • The equation for the perpendicular bisector must be:
    • x=(6)+02=3x = \frac{(-6)+0}{2} = -3
Illustrative background for Find the centre of the circumcircleIllustrative background for Find the centre of the circumcircle ?? "content

Find the centre of the circumcircle

  • The centre of the circumcircle is the point of intersection between the perpendicular bisectors of AB and AC.
Illustrative background for Find the centre of the circumcircleIllustrative background for Find the centre of the circumcircle ?? "content

Find the centre of the circumcircle

  • Substituting in x=3x=3 into y=3x+16y = 3x +16, we get:
    • y=(3×3)+16y=7y = (3 \times 3) +16 \rightarrow y = 7
  • The centre of the circumcircle has coordinates (3,7).
Illustrative background for Find the radius of the circumcircleIllustrative background for Find the radius of the circumcircle ?? "content

Find the radius of the circumcircle

  • The radius of the circumcircle is equal to the distance between the centre and any one of the points of the triangle.
Illustrative background for Find the radius of the circumcircleIllustrative background for Find the radius of the circumcircle ?? "content

Find the radius of the circumcircle

  • The distance between the centre and point AA is found by using Pythagoras' theorem:
    • Distance =(3(6))2+(73)2=5= \sqrt{(3-(-6))^2 + (7-3)^2} = 5
  • The radius of the circumcircle is equal to 5.
Illustrative background for Write the equation of the circumcircleIllustrative background for Write the equation of the circumcircle ?? "content

Write the equation of the circumcircle

  • We can now write the equation of the circumcircle in its general form:
    • (x+3)2+(y7)2=25(x+3)^2 + (y-7)^2 = 25

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

Go student ad image

Unlock your full potential with GoStudent tutoring

  • Affordable 1:1 tutoring from the comfort of your home

  • Tutors are matched to your specific learning needs

  • 30+ school subjects covered

Book a free trial lesson