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.  y x 180 – 2x C x B 180 – 2y 2x+2y = 2(x+y)  y A

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

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

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

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

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.