Sec 3 E-Math Properties of Circles topical questions

O is the centre of a circle and angle AOB = 76° at the centre. C lies on the major arc. Find angle ACB, with a reason.

An angle at the circumference standing on arc PQ is 42°. Find the angle at the centre standing on PQ.

AB is a diameter of a circle and C lies on the circle with angle CAB = 34°. Find angle ABC.

PQRS is a cyclic quadrilateral with angle P = 98° and angle Q = 85°. Find angles R and S.

Points X and Y lie in the same segment of a circle, standing on chord PQ. Angle PXQ = 55°. Write down angle PYQ with a reason.

O is the centre of a circle and A, B, C lie on it, with C on the major arc AB. Given angle AOB = 140°, find angle ACB and name the property used.

From an external point P, tangents touch a circle at A and B, with PA = 7.4 cm. State the length of PB and the size of angle OAP, each with a reason.

TC is a tangent to a circle at C, with O the centre. The angle between the tangent TC and the chord CQ is 58°. Find angle OCQ.

A chord of length 16 cm lies in a circle of radius 10 cm. Find the distance from the centre of the circle to the chord.

A chord AB is 24 cm long and sits 5 cm from the centre O of its circle. Find the radius of the circle.

From external point P, tangents PA and PB touch a circle with centre O at A and B. Given angle APB = 50°, find angle AOB.

ABCD is a cyclic quadrilateral in which angle A = x° and the opposite angle C = (3x − 20)°. (a) Form an equation in x. (b) Find angles A and C.

A circle has centre O and radius 10 cm. A chord AB measures 16 cm. (a) Find the distance from O to AB. (b) A second chord CD, drawn elsewhere in the same circle, also measures 16 cm. State the distance from O to CD and justify your answer.

1. (a)Distance from O to AB.[3]
2. (b)Distance from O to CD, with reason.[2]

PQRS is a cyclic quadrilateral with angle P = 3x°, angle R = (2x + 25)° and angle Q = 110°. (a) Form an equation in x and solve it. (b) State angles P and R. (c) Find angle S.

1. (a)Form and solve the equation.[2]
2. (b)Angles P and R.[1]
3. (c)Angle S.[1]

AB is a diameter of a circle with centre O, and C and D are points on the circle such that angle CAB = 28°. (a) Find angle ABC. (b) Find angle ACB. (c) Find angle BOC, where O is the centre.

1. (a)Angle ABC.[2]
2. (b)Angle ACB.[1]
3. (c)Angle BOC.[2]

A circle has two parallel chords, each 10 cm long. One lies 12 cm from the centre. (a) Explain why the other chord also lies 12 cm from the centre. (b) Find the radius of the circle.

1. (a)Why 12 cm for both.[2]
2. (b)Radius.[3]

TA is a tangent at A to a circle with centre O, and B is a point on the circle with angle TAB = 38°. (a) Find angle OAB. (b) Explain why triangle OAB is isosceles and find angle OBA. (c) Find angle AOB and the angle ACB subtended by AB at a point C on the major arc.

1. (a)Angle OAB.[2]
2. (b)Isosceles reason and angle OBA.[2]
3. (c)Angle AOB and angle ACB.[2]

A, B, C and D lie on a circle with AB a diameter. Angle DAB = 55°, with D on the semicircle opposite C. (a) Find angle ADB. (b) Find angle ABD. (c) Find angle DCB.

1. (a)Angle ADB.[2]
2. (b)Angle ABD.[2]
3. (c)Angle DCB.[2]

Two concentric circles have radii 10 cm and 6 cm. A chord of the larger circle is tangent to the smaller circle. (a) Explain why the radius of the smaller circle drawn to the contact point is perpendicular to the chord. (b) Find the length of the chord.

1. (a)Explain the right angle.[2]
2. (b)Find the chord length.[4]

From an external point P, tangents PA and PB touch a circle at A and B, and angle APB = 64°. C is a point on the major arc AB. Find (a) angle AOB (O the centre), (b) angle ACB, and (c) angle PAB.

1. (a)Angle AOB.[2]
2. (b)Angle ACB.[2]
3. (c)Angle PAB.[2]