3.2.9

Circumcircle of a Right-angled Triangle

Test yourself

Circumcircle of a Right-Angled Triangle

The circumcircle of a right-angled triangle is easily found by looking at the midpoint of its hypotenuse.

Illustrative background for Triangle in a semi-circleIllustrative background for Triangle in a semi-circle ?? "content

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.
Illustrative background for How do you find the hypotenuse of the right-angled triangle?Illustrative background for How do you find the hypotenuse of the right-angled triangle? ?? "content

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
Illustrative background for How do you find the equation of the circumcircle?Illustrative background for How do you find the equation of the circumcircle? ?? "content

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.
Illustrative background for ExampleIllustrative background for Example ?? "content

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.
Illustrative background for Work out the length of each side squaredIllustrative background for Work out the length of each side squared ?? "content

Work out the length of each side squared

  • AB2=22=4AB^2 = 2^2 = 4
  • AC2=12+12=2AC^2 = 1^2 + 1^2 = 2
  • BC2=12+12=2BC^2 = 1^2 + 1^2 = 2
Illustrative background for Use Pythagoras' theoremIllustrative background for Use Pythagoras' theorem ?? "content

Use Pythagoras' theorem

  • All right-angled triangles satisfy Pythagoras' theorem.
  • We can see that AC2+BC2=AB2AC^2 + BC^2=AB^2 as 2 + 2 = 4.
  • This means that ABCABC is a right-angled triangle with ABAB as its hypotenuse.
Illustrative background for Use circle theoremIllustrative background for Use circle theorem ?? "content

Use circle theorem

  • ABCABC is a right-angled triangle, so ABAB is the diameter of its circumcircle.
Illustrative background for Find the centreIllustrative background for Find the centre ?? "content

Find the centre

  • The centre of the circumcircle is the midpoint of the diameter ABAB.
    • Midpoint = (3+32,6+42)=(3,5)(\frac{3+3}{2},\frac{6+4}{2}) = (3,5)
Illustrative background for Find the radiusIllustrative background for Find the radius ?? "content

Find the radius

  • As calculated earlier, AB2AB^2 = 4. This means:
    • Diameter ABAB = 2
    • Radius = 1
Illustrative background for Write the equation of the circleIllustrative background for Write the equation of the circle ?? "content

Write the equation of the circle

  • We can now write the equation of the circumcircle in its general form:
    • (x3)2+(y5)2=1(x-3)^2 + (y-5)^2 = 1

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