Circle Theorems Revision

1. Circle Theorems Revision

2. C O B A Circle Theorem 1 • The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. AOB = 2x ACB

3. C B D A Circle Theorem 2 • Every angle at the circumference of a semicircle that is subtended by the diameter of the semicircle is a right angle.

4. C2 C3 C1 B A Circle Theorem 3 • Angles at the circumference in the same segment of a circle are equal. Points on the circumference are subtended by the same arc AB. • AC1B = AC2B

5. D C B A Circle Theorem 4 • The sum of the opposite angles of a cyclic quadrilateral is 180o.

6. O A X B Circle Theorem 5 • A tangent to a circle is perpendicular to the radius drawn to the point of contact. • OX is perpendicular to AB

7. X Y A Circle Theorem 6 • Tangents to a circle from an external point to the point of contact are equal in length. • AX = AY

8. X O Y A Circle Theorem 7 • The line joining an external point to the centre of the circle bisects the angle between the tangents. • OAX = OAY

9. O B M C Circle Theorem 8 • A radius bisects a chord at 90o. • If O is the centre of the circle BMO = 90o, BM = CM

10. B A P Q T Circle Theorem 9 • The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. PTA = TBA

11. C D C C2 B B X X C3 O A C1 C O O O B A Y Y M B B B D A A X A A A A P Q C T B 1 2 3 4 6 7 5 8 9