Circle theorems

A, B and C are points on a circle, centre O.

Prove that y = 180 – 5 x

A, B, C and D are points on a circle. EBF is a tangent.

Work out the value of x.

A, B, C and D are points on a circle, centre O. angle OBC: angle ODC = 5: 3

Work out the size of angle OBC. You must show your working.

A, B and C are points on a circle. PAC is a straight line. PB is a tangent to the circle.

P, Q and R are points on a circle, centre O. Angle POR = 120 ° PQ = 4 cm QR = 5 cm

Work out the radius of the circle. Give your answer in the form where k is an integer.

A, B, C, D and E are points on a circle, centre O.

Work out the value of x.

B, C and D are points on a circle, centre P. AB and AC are tangents to the circle.

PQRS is a cyclic quadrilateral.

Angle PSR = 4(x + 15 °) Angle PQR is 40 ° smaller than angle PSR.

Work out the value of x.

F, H, K and J are points on a circle. Chords HJ and KF intersect at L. EFG is a tangent to the circle. FH and JK are parallel.

Angle FHJ = 2 x

Give reasons why angle FKJ and angle HJK are also equal to 2 x.

Work out the values of x and y. You must show your working. Do not use trial and improvement.

A, B and C are points on a circle, centre O. ACD is a straight line. Angle BCD = w

Prove that w = x + 90 °