4.5.3

Angle of a Semicircle

Test yourself

Circle Theorems

Illustrative background for ProofIllustrative background for Proof ?? "content

Proof

  • Split the triangle into two triangles which are both isosceles since they both have two sides which are radii.
  • Mark one of the angles at the centre x.
Illustrative background for Proof continuedIllustrative background for Proof continued ?? "content

Proof continued

  • y = 12(180° - x) since the triangle is isosceles and all angles add up to 180°.
  • Similarly z = 12(180° - (180° - x)) = 12x
  • Therefore the angle at the circumference is z + y = 12 × 180° = 90° as required.
Illustrative background for Alternative proofIllustrative background for Alternative proof ?? "content

Alternative proof

  • Alternatively, using the previous theorem we see that the angle at the centre is twice the angle at the circumference.
  • So 180° is twice the angle at the circumference so the angle is 90°.

Jump to other topics

1Number

1.1Using Numbers

1.2Fractions, Decimals & Percentages

1.3Powers & Roots

1.4Accuracy

2Algebra

2.1Introduction to Algebra

2.2Manipulating Algebra

2.3Proofs & Functions

2.4Straight Line Graphs

2.5Common Graphs

2.6Transformations & Tangents

2.7Properties of Graphs

2.8Solving Equations

2.9Inequalities

2.10Sequences

3Ratio

4Geometry

4.1Introduction to Geometry

4.2Triangles & Quadrilaterals

4.3Transformations

4.4Circle Basics

4.5Circle Theorems

4.6Measurements & Units

4.7Calculating Area

4.8Triangle Formulae

4.93D Shapes

4.10Vectors

5Probability

6Statistics

Go student ad image

Unlock your full potential with GoStudent tutoring

  • Affordable 1:1 tutoring from the comfort of your home

  • Tutors are matched to your specific learning needs

  • 30+ school subjects covered

Book a free trial lesson