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**Proofs for circle theorems**

Tuesday, 13 July 2010

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**1. Angle subtended at the centre**

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

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**2. Angles subtended from the same arc.**

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

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**3. Angles in a semi-circle.**

The largest angle in a semi-circle will always be 90 90 C A B

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**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 Assume that the tangent is not perpendicular to the radius. Now there must be a perpendicular from the center of the circle to the tangent. It must intersect the tangent elsewhere (apart from the point on intersection). Now since a perpendicular is the shortest distance of a point from a line, the perpendicular must have distance < radius of the circle (since line from point of intersection of tangent to center of circle is a radius). => the foot of the perpendicular must lie within the circle. =>the line meets the circle elsewhere and the line is not really a tangent! Thus there is a contradiction. Therefore the radius is the shortest distance. Consequently, it must be perpendicular to the tangent at the point of intersection 90 A The angle between a tangent an its radius will always be 90

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**5. Angles in a cyclic quadrilateral.**

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

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**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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