Circles quiz

A tangent to a circle intersects it in:

The angle between a tangent to a circle and the radius drawn through the point of contact is:

A line intersecting a circle in two points is called a:

The length of the tangent from a point A at a distance of 5 cm from the centre of a circle of radius 3 cm is:

The number of tangents that can be drawn to a circle from a point inside it is:

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at 70 degrees, then angle AOB equals:

The length of the tangent from a point 13 cm away from the centre of a circle of radius 5 cm is:

Tangents drawn at the ends of a diameter of a circle are:

A quadrilateral ABCD circumscribes a circle with AB = 6 cm, BC = 7 cm and CD = 4 cm. The length of AD is:

The maximum number of tangents that can be drawn to a circle parallel to a given line is:

Prove that the lengths of tangents drawn from an external point to a circle are equal. (2 marks)

Explain why there is one and only one tangent at any point of a circle. (2 marks)

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. (2 marks)

The length of a tangent from a point A at a distance of 5 cm from the centre of a circle is 4 cm. Find the radius of the circle. (2 marks)

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre. (2 marks)

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ. (2 marks)

If two tangents inclined at an angle of 60 degrees are drawn to a circle of radius 3 cm, find the length of each tangent. (2 marks)

Prove that the centre of a circle touching two intersecting lines lies on the bisector of the angle between the lines. (2 marks)

A circle touches all four sides of a quadrilateral ABCD with AB = 9 cm, CD = 7 cm and BC = 10 cm. Find AD. (2 marks)

Explain the difference between a secant and a tangent using the idea of a secant moving away from the centre until its two points of intersection coincide. (2 marks)

Two concentric circles have radii 13 cm and 5 cm. Find the length of the chord of the larger circle which touches the smaller circle. (3 marks)

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. (3 marks)

From an external point P, two tangents PA and PB are drawn to a circle with centre O and radius 6 cm. If OP = 10 cm, find the length of each tangent and the area of the quadrilateral OAPB. (3 marks)

The radius of the incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and 8 cm. A student says the triangle is right-angled. Check this claim by finding all the sides. (3 marks)

In a right triangle ABC, right-angled at B, AB = 24 cm and BC = 7 cm. Find the radius of the circle inscribed in the triangle. (3 marks)

Read the passage and answer the questions. A circular park of radius 20 m has a straight road touching its boundary at point P. A lamp post L stands on the road at a distance of 21 m from P. The park gardener walks from the lamp post straight to the fountain at the centre O. (i) What is the measure of angle OPL? Give the reason. (ii) Find the distance OL walked by the gardener. (iii) If another straight path from L touches the park at Q, how long is LQ? Name the theorem used. (5 marks)

Read the passage and answer the questions. A bicycle chain runs around a circular gear. Two straight parts of the chain leave the gear at points A and B and meet at a point P outside the gear, forming an angle APB of 60 degrees. The gear has radius 7 cm and centre O. (i) What are PA and PB called with respect to the gear, and how are their lengths related? (ii) Find angle AOB. (iii) Find OP and the length PA. (Use sqrt(3) = 1.73) (5 marks)

Read the passage and answer the questions. An artist designs a logo in which a circle is inscribed in a triangle ABC. The circle touches BC at D, CA at E and AB at F. The artist measures AF = 4 cm, BD = 6 cm and CE = 5 cm. (i) Write the lengths AE, BF and CD, giving a reason. (ii) Find the perimeter of triangle ABC. (iii) Find the lengths of the three sides of the triangle. (5 marks)

Read the passage and answer the questions. Two roads touch a circular roundabout of radius 12 m at points A and B and meet at a junction J. Traffic engineers find that JA = 16 m. The centre of the roundabout is O. (i) What is the length JB? Why? (ii) Find the distance OJ. (iii) Find the area of the quadrilateral OAJB. (5 marks)

Read the passage and answer the questions. A window frame is in the shape of a square ABCD whose four sides all touch a circular glass pane. The side of the square is 40 cm. (i) What is the radius of the circular pane? Give a reason. (ii) Verify that AB + CD = AD + BC for this frame. (iii) If the frame were a rhombus of side 40 cm circumscribing a circle, would the relation in (ii) still hold? Explain. (5 marks)

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Draw a neat labelled figure and explain why every point on the tangent, other than the point of contact, lies outside the circle. (6 marks)

Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this result, prove that if a circle is inscribed in a triangle ABC touching BC, CA and AB at D, E and F, then AF + BD + CE = AE + BF + CD = half the perimeter of the triangle. Draw a labelled figure. (6 marks)

In the figure, XY and X'Y' are two parallel tangents to a circle with centre O, and another tangent AB with point of contact C intersects XY at A and X'Y' at B. Draw the figure and prove that angle AOB = 90 degrees. (6 marks)

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Draw a neat labelled figure. (6 marks)

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are 8 cm and 6 cm. Draw the figure, form an equation using the area of triangle ABC in two ways (by Heron's formula and as the sum of areas of triangles OBC, OCA and OAB), and find AB and AC. (6 marks)