Circle Theorems - Mathematics

  • Created by: v.a..
  • Created on: 17-04-19 19:08

Perpendicular bisector goes through centre

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

  • Draw the perpendicular from centre, O, to the chord AB.
  • Draw a radius to A and another to B to form two isosceles triangles.
  • 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.
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Centre angle is double circumference angle

Angle at centre is double the angle at the circumference

  • 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.
  • 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.
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Angle in a semicircle is 90°

Angle in a semicircle is 90°

  • 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.
  • 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.
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Alternative proof for 90° angle in semicircle

  • 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°.
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Opposite angles add to 180°

  • Opposite angles in a cyclic quadrilateral add up to 180°.
    • Here x + y = 180°
  • Label the opposite angles, then draw two radii from the other corners.
  • Use ‘angle at centre twice that at the circumference’ to get the angles at the centre as 2x, 2y.
  • Since angles at a point add up to 360°:
    • 2x + 2y = 360° and so x + y = 180°.
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Lengths of tangents

Two tangents to the same circle from the same point (C) will be the same length. Here AC = BC

  • These are congruent triangles (by using the 'RHS' rule).
  • They share a common side (length x) and another side which is a radius.
  • Therefore the remaining side, the tangent, is the same length for both.
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Angles in the same segment are equal

All triangles drawn from a chord in the same segment will have the same angle at the circumference.

  • Draw in radii from the ends of the chord.
  • Label the angle at the centre.
  • Using ‘angle at centre twice that at the circumference’ we see that 
  • z = 2x, z = 2y and so x = y.
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Alternate segment theorem

The angle between a tangent and a chord is always equal to the angle at the circumference of any triangle formed from the chord.

  • Draw in a diameter from the point where the tangent touches. Connect it to the other end of the chord.
  • Using ‘radius and tangent meet at a right angle’ we get a right angle where the diameter meets the tangent.
  • Using ‘angle in a semicircle is 90°’ we get that the angle at the other end of the chord is a right angle too.
  • We find that the angle at the bottom of the diameter is the same as x.
  • Using ‘angles in same segment’ we get that x = y.
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