3.2.6

Circle Theorems - Angle at a Semi-Circle

Test yourself

Circle Theorems

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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.
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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.
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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

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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