# 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 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 #### Exam Tip

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

#### Worked Example  