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Circle Theorems

Maths: Edexcel IGCSE Higher, topic 4.5.2

Circle Theorems

Perpendicular bisector goes through centre

Perpendicular bisector goes through centre

  • The perpendicular bisector of a chord always goes through the centre of the circle.
Proof

Proof

  • Draw the perpendicular from centre, O, to the chord AB.
  • Draw a radius to A and another to B to form two triangles.
Proof continued

Proof continued

  • Their hypotenuses are the same length (because they are both radii) and they share another edge so the triangles are congruent by the ‘RHS’ rule.
  • Therefore AM = BM and so the perpendicular splits the chord exactly in half.

Circle Theorems

Angle at centre is double the angle at the circumference

Angle at centre is double the angle at the circumference

Proof

Proof

  • Split the shape into two isosceles triangles and label the angles.
  • The angle at the centre is 360° - x - y and the angle at the circumference is a + b.
Proof continued

Proof continued

  • Since the angles in a triangle add up to 180°:
    a = 1⁄2(180° - y), b = 1⁄2(180° - x)
  • So angle at circumference = a + b = 1⁄2(360° - x - y)
    = 1⁄2 × angle at centre.

Circle Theorems

Angle in a semicircle is 90°

Angle in a semicircle is 90°

Proof

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.
Proof continued

Proof continued

  • y = 1⁄2(180° - x) since the triangle is isosceles and all angles add up to 180°.
  • Similarly z = 1⁄2(180° - (180° - x)) = 1⁄2x
  • Therefore the angle at the circumference is z + y = 1⁄2 × 180° = 90° as required.
Alternative proof

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°.

Test yourself on Circle Theorems

  • The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?

  • The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?

  • Which two lines are the same length?

  • What is the value of angle x?

  • What is the value of angle x?

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