Angle in a Semicircle (Cambridge International AS Mathematics)

Angle in a semicircle

What are angles in a semicircle?

• 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 circumcircle
• It also says that any angle at the circumference in a semicircle is a right angle

How can I use angles in a semicircle 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 Hint

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

Worked Example