Circles quiz

The longest chord of a circle is its:

The perpendicular from the centre of a circle to a chord:

The angle in a semicircle is:

If an arc subtends 80 degrees at the centre, the angle it subtends at a point on the remaining part of the circle is:

The sum of a pair of opposite angles of a cyclic quadrilateral is:

Equal chords of a circle are:

Circles having the same centre but different radii are called:

Angles in the same segment of a circle are:

A chord of a circle of radius 5 cm is at a distance of 3 cm from the centre. Its length is:

In a cyclic quadrilateral PQRS, angle P = 100 degrees. Then angle R is:

Prove that the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. (2 marks)

If two chords of a circle subtend equal angles at the centre, show that they are equal. (2 marks)

In a circle with centre O, angle AOB = 120 degrees and C is a point on the major arc. Find angle ACB. (2 marks)

PQRS is a cyclic quadrilateral with angle Q = 3x and angle S = 2x. Find x and the two angles. (2 marks)

Two chords AB and CD are each 8 cm long in a circle of radius 5 cm. Find the distance of each from the centre. (2 marks)

Show that a cyclic parallelogram is a rectangle. (2 marks)

In a circle with centre O, AB is a diameter and C is a point on the circle with angle CAB = 35 degrees. Find angle ABC. (2 marks)

Why can a circle not be drawn through three collinear points? (2 marks)

Angle ACB = 50 degrees is an angle in a segment of a circle. Find the angle subtended by the chord AB at the centre. (2 marks)

Prove that if two circles intersect at two points, the line through their centres is the perpendicular bisector of the common chord. (2 marks)

A chord is 16 cm long and lies 6 cm from the centre of a circle. Find the radius of the circle. (3 marks)

Two parallel chords AB = 10 cm and CD = 24 cm of a circle lie on the same side of the centre. If the radius is 13 cm, find the distance between them. (3 marks)

ABCD is a cyclic quadrilateral in which angle A = (2x + 10) degrees and angle C = (3x + 20) degrees. Find x and angles A and C. (3 marks)

In a circle with centre O, chord PQ subtends angle POQ = 100 degrees. Find angle PRQ where R is on the major arc and angle PSQ where S is on the minor arc. (3 marks)

In a circle of radius 17 cm, a chord is at a distance of 8 cm from the centre. Find the length of the chord. If another chord of the same circle is 16 cm long, find its distance from the centre. (3 marks)

Read the passage and answer the questions. Three friends stand on a circular park boundary of radius 25 m. Two of them, P and Q, are 48 m apart and the centre O is a fountain. (i) Find the distance of the fountain from the line PQ. (ii) If the third friend R also stands on the boundary, what can you say about angle POQ compared with angle PRQ, when R is on the major arc? (iii) If angle PRQ is 37 degrees, find angle POQ. (5 marks)

Read the passage and answer the questions. A circular clock face of radius 10 cm has a straight metal strip fixed as a chord 12 cm long. (i) Find the distance of the strip from the centre. (ii) Another strip of the same length is fixed elsewhere. How far is it from the centre? Give the theorem. (iii) What is the length of the longest strip that can be fixed on this clock face? (5 marks)

Read the passage and answer the questions. A garden is shaped as a cyclic quadrilateral ABCD. The angle at corner A is 85 degrees and at corner B is 95 degrees. (i) Find angle C. (ii) Find angle D. (iii) Show that AD is parallel to BC and that AB is not parallel to DC. (5 marks)

Read the passage and answer the questions. A semicircular window has its straight edge AB as a diameter of 1.2 m. A decorative rod joins A and B to a point C on the curved edge. (i) What is angle ACB? (ii) If AC = 0.72 m, find BC. (iii) State the theorem used in part (i). (5 marks)

Read the passage and answer the questions. A circular wheel has two spokes OA and OB with angle AOB = 70 degrees. A point P on the rim lies on the major arc AB, and a point Q on the rim lies on the minor arc AB. (i) Find angle APB. (ii) Find angle AQB. (iii) What is the relation between angles APB and AQB? (5 marks)

Prove that equal chords of a circle are equidistant from the centre. Draw a neat figure and give reasons for every step. (6 marks)

Prove that the angle subtended by an arc at the centre is double the angle at any point on the remaining part of the circle. Use it to prove that angles in the same segment are equal. (6 marks)

Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees. Use it to find all angles of a cyclic quadrilateral PQRS where angle P = 70 degrees and angle Q = 105 degrees. (6 marks)

Draw a circle of radius 5 cm and two parallel chords of lengths 6 cm and 8 cm. Find the distance between them when they are (i) on the same side of the centre (ii) on opposite sides of the centre. (6 marks)

Prove that if the sum of a pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic. Hence show that an isosceles trapezium is cyclic. (6 marks)