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GCSE Mathematics (1-9) 2015 How To Do it G10Theory Geometry - Circle theorems - 10 apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results 1 Angle at the centre The angle at the centre is twice the angle at the circumference 1.1 Proof See second diagram AB = BC (both radius of circle) So triangle ACB is isosceles So angle CAB = angle BCA = w angle ABC = 180 -2y (angle sum in triangle is 180) Similarly angle ABD = 180 - 2z so CBD = ABC + ABD = 360- 2(y+z) so w = 360-CBD = 2(y+z) (angle sum at a point = 360 ) GCSE Mathematics (1-9) 2015 2 How To Do it Angle on a diameter The angle on a diameter is 90 degrees 2.1 Proof This follows from 'angle at the centre' If the angle at the centre = 180 (straight line) then the angle at the circumference is 90 3 Angles in the same segment Angle ABC = angle ADC 3.1 Proof AOC = 2 ABC ( angle at centre ) AOC = 2 ADC ( angle at centre ) So ABC = ADC 4 Cyclic quadrilateral A cyclic quadrilateral has its vertices on a circle, as shown Opposite angles add up to 180 a+b=180 4.1 Proof See second diagram AO = BO ( both radius of circle) So triangle AOB is isosceles. So angle OAB = angle ABO and similarly for the other vertices. G10Theory GCSE Mathematics (1-9) 2015 How To Do it G10Theory Angles in a quadrilateral add up to 360 so A x w 2x+2y+2z+2w = 360 B so (x+w) + (y+z) = 180 x y O so DAB = DCB y z also (x+y) + (w+z) = 180 so ABC + ADC = 180 5 Tangent is perpendicular to radius If BC is a tangent, angle BAO is a right angle. 5.1 Proof Suppose its not, and that instead OE is perpendicular to the tangent BC. The perpendicular is the shortest distance to a point. So OE < OA So OE < radius But OE = OD + DE = radius + DE So OE < radius is impossible So the initial assumption is false C w z D GCSE Mathematics (1-9) 2015 6 How To Do it G10Theory Two tangents equal AB = CB 6.1 Proof Triangles OAB and OCB are congruent ( RHS - the side is the radius and the hypotenuse is common ) So AB = CB 7 Alternate segment a=b The line is a tangent to the circle. It is the 'alternate' segment because a is in the other segment from b : the alternate segment. 7.1 Proof OB = OC (both radius) so OBC is isosceles so CBO = OCB and similarly for the other triangles OAD is a right angle (tangent is perpendicular) So CAD = 90 - z 2x+2y+2z=180 (angles in a triangle) x+y+z=90 x+y=90-z so CAD = CBA a b GCSE Mathematics (1-9) 2015 8 How To Do it Perpendicular bisects chord AB = BC 8.1 Proof Triangles OBC and OBA are congruent ( RHS : hypotenuse is radius, side is common ) So BC = AB G10Theory