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Proofs for circle theorems

Proofs for circle theorems. Tuesday, 13 July 2010. 1. Angle subtended at the centre. The angle subtended at the centre from an arc is double the angle at the circumference. . x. y. 180 – 2x. x. C. B. 180 – 2y. 2x+2y = 2( x+y ). . y. A. 2 . Angles subtended from the same arc.

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Proofs for circle theorems

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  1. Proofs for circle theorems Tuesday, 13 July 2010

  2. 1. Angle subtended at the centre The angle subtended at the centre from an arc is double the angle at the circumference.  x y 180 – 2x x C B 180 – 2y 2x+2y = 2(x+y)  y A

  3. 2. Angles subtended from the same arc. Angles subtended from the same arc are equal.   C B A

  4. 3. Angles in a semi-circle. The largest angle in a semi-circle will always be 90 90 C A B

  5. 4. The Angle between a Tangent and its radius. Definition: A tangent is a line that will touch the circle at one point only. (i.e. it does not cut the circle) C Tangent 90 A The angle between a tangent an its radius will always be 90

  6. 5. Angles in a cyclic quadrilateral. a Definition: A cyclic quadrilateral is any four-sided polygon whose four corners touch the circumference of the circle 180 –  d b 2 360 –2 Opposite angles in a cyclic quadrilateral add up to 180  c

  7. 4. The Angle between a Tangent and a chord. Definition: A chord is any straight line which touches the circumference at two points. The largest chord possible is called the diameter. =180 – 90 – (90 – ) =   Tangent 90 – 90  Chord The angle between a tangent a chord is equal to the angle in the alternate segment.

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