AQA A Level Maths: Pure复习笔记3.2.4 Angle in a Semicircle

Angle in a Semicircle


What is the angle in a semicircle property?

  • It is always possible to draw a unique circle through the three vertices of a triangle – this is called the circumcircle of the triangle


  • The angle in a semicircle property says that If a triangle is right-angled, then its hypotenuse is a diameter of its circumcircle3.2.4-Angle-in-Semic-Illustr-2
  • It also says that any angle at the circumference in a semicircle is a right angle

How can I use the angle in a semicircle property to find the equation of a circle?

  • Because the hypotenuse of a right-angled triangle is a diameter of the triangle's circumcircle you also know that:
    1. the radius of the circumcircle is half the length of the hypotenuse
    2. the centre of the circumcircle is the midpoint of the hypotenuse



  • Once you know the radius and the centre you can write down the equation of the circle


Exam Tip

  • To show that a triangle is right-angled, show that the lengths of its sides satisfy Pythagoras' theorem.

Worked Example