Circle theorems

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Circle theorems

  1. 1. Circle Theorems
  2. 2. A Circle features……. Circumference … the distance around the Circle… … its PERIMETER … the distance across the circle, passing through the centre of the circle … the distance from the centre of the circle to any point on the circumference Diameter Radius
  3. 3. A Circle features……. … a line joining two points on the circumference. … chord divides circle into two segments … part of the circumference of a circle Major Segment Minor Segment … a line which touches the circumference at one point only From Italian tangere , to touch Chord Tangent ARC
  4. 4. Properties of circles <ul><li>When angles, triangles and quadrilaterals are constructed in a circle, the angles have certain properties </li></ul><ul><li>We are going to look at 4 such properties before trying out some questions together </li></ul>
  5. 5. An ANGLE on a chord An angle that ‘sits’ on a chord does not change as the APEX moves around the circumference … as long as it stays in the same segment We say “Angles subtended by a chord in the same segment are equal” Alternatively “Angles subtended by an arc in the same segment are equal” From now on, we will only consider the CHORD, not the ARC
  6. 6. Typical examples Find angles a and b Angle b = 28º Angle a = 44º Very often, the exam tries to confuse you by drawing in the chords YOU have to see the Angles on the same chord for yourself Imagine the Chord Imagine the Chord
  7. 7. Angle at the centre Consider the two angles which stand on this same chord Chord It is half the angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” A What do you notice about the angle at the circumference?
  8. 8. Angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” It’s still true when we move The apex, A, around the circumference As long as it stays in the same segment Of course, the reflex angle at the centre is twice the angle at circumference too!! A 136° 272°
  9. 9. Angle at Centre A Special Case When the angle stands on the diameter, what is the size of angle a? The diameter is a straight line so the angle at the centre is 180° Angle a = 90° We say “The angle in a semi-circle is a Right Angle” a a
  10. 10. A Cyclic Quadrilateral … is a Quadrilateral whose vertices lie on the circumference of a circle Opposite angles in a Cyclic Quadrilateral Add up to 180° They are supplementary We say “ Opposite angles in a cyclic quadrilateral add up to 180°”
  11. 11. Questions
  12. 15. Could you define a rule for this situation?
  13. 16. Tangents <ul><li>When a tangent to a circle is drawn, the angles inside & outside the circle have several properties. </li></ul>
  14. 17. 1. Tangent & Radius A tangent is perpendicular to the radius of a circle
  15. 18. 2. Two tangents from a point outside circle PA = PB Tangents are equal PO bisects angle APB <PAO = <PBO = 90° 90° 90° <APO = <BPO AO = BO (Radii) The two Triangles APO and BPO are Congruent g g
  16. 19. 3 Alternate Segment Theorem The angle between a tangent and a chord is equal to any Angle in the alternate segment Angle between tangent & chord Alternate Segment Angle in Alternate Segment We say “ The angle between a tangent and a chord is equal to any Angle in the alternate (opposite) segment”

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