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ASMaths: PureAQARevision Notes3. Coordinate Geometry3.2 Circles3.2.4 Angle in a Semicircle

Angle in a Semicircle (AQA AS Maths: Pure)

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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

 Circumcircle, A Level & AS Level Pure Maths Revision Notes

  • The angle in a semicircle property says that If a triangle is right-angled, then its hypotenuse is a diameter of its circumcircleAngle in Semic Illustr 2, A Level & AS Level Pure Maths Revision Notes
  • 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

Angle in Semic Illustr 3, A Level & AS Level Pure Maths Revision Notes

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

Angle in Semic Illustr 4, A Level & AS Level Pure Maths Revision Notes

Examiner Tip

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

Worked example

Angle in Semic Example, A Level & AS Level Pure Maths Revision Notes

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    Roger

    Author: Roger

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.