Angle in a Semicircle (OCR AS Maths A): Revision Note

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 circumcircle

  

• 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