Circles

In earlier classes you studied circles, chords and arcs. In this chapter you will study the different ways a line can be placed relative to a circle: it may not meet the circle at all, it may cut it at two points as a secant, or it may touch it at exactly one point as a tangent. The point where a tangent touches the circle is called the point of contact. Tangents appear in real life whenever a wheel rolls on a road or a belt runs over a pulley. You will prove two important theorems. The first states that the tangent at any point of a circle is perpendicular to the radius through the point of contact, and that there is exactly one tangent at a point on the circle. The second states that the lengths of the two tangents drawn from an external point to a circle are equal. Using these results with Pythagoras theorem and properties of triangles, you will solve problems on tangent lengths, concentric circles and quadrilaterals circumscribing a circle.

Detailed Worksheet: Circles Section A - Definitions (10 marks) 1. Define a tangent to a circle and a secant to a circle. What is the point of contact? (2 marks) 2. How many tangents can be drawn to a circle from a point (i) inside the circle (ii) on the circle (iii) outside the circle? (2 marks) 3. State the theorem relating the tangent at a point of a circle and the radius through the point of contact. (2 marks) 4. Define the length of a tangent from an external point. State the theorem about the lengths of tangents drawn from an external point. (2 marks) 5. What is meant by a quadrilateral circumscribing a circle? State the relation between its opposite sides. (2 marks) Section B - Calculations and Applications (15 marks) 6. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle. (3 marks) 7. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. (3 marks) 8. A circle is inscribed in a triangle ABC with AB = 10 cm, BC = 8 cm and CA = 12 cm. Find the lengths of the tangents from A, B and C to the circle. (3 marks) 9. Two tangents PA and PB are drawn to a circle with centre O from an external point P. If angle APB = 80 degrees, find angle AOB and angle POA. (3 marks) 10. A circle is inscribed in a right triangle with legs 6 cm and 8 cm. Using the property of tangents from an external point, find the radius of the circle. (3 marks) Section C - Diagrams (10 marks) 11. Draw a circle with centre O and a line AB touching it at P. Mark the radius OP and a point Q on AB other than P. Use the figure to explain why OP is the shortest distance from O to AB, and hence why OP is perpendicular to AB. (4 marks) 12. Draw a labelled figure showing two tangents PA and PB drawn from an external point P to a circle with centre O. Join OA, OB and OP and name the pair of congruent triangles that proves PA = PB. (3 marks) 13. Draw a figure of two concentric circles with centre O in which a chord AB of the larger circle touches the smaller circle at P. Show that P is the midpoint of AB. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. Hence explain why a circle cannot have more than two tangents parallel to a given line. (5 marks) 15. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC. Hence show that a parallelogram circumscribing a circle must be a rhombus. (5 marks) 16. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP, showing each step clearly. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat figures for every proof and state the theorem or reason used at each step.