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Year 10 Maths — Circles and tangents

Use the two tangent properties of a circle: a tangent meets the radius at a right angle, and the two tangents from an outside point are equal.

Name: ________________________

Before you start

Three straight lines have been drawn on this circle. Find out what each one is called.

123
  1. 1. Diameter
  2. 2. Chord
  3. 3. Tangent

The tangent is the only one of the three that never gets inside the circle. And a diameter is not a separate kind of thing at all — it is simply a chord that happens to pass through the centre.

  1. 1.A tangent from a point P touches a circle with centre O at A. The radius OA is 8 cm and the hypotenuse OP is 17 cm. How long is the tangent PA, in centimetres?

  2. 2.Two tangents are drawn from a point P to a circle, touching it at A and B. How does PB compare with PA?

    • a) PB is twice PA
    • b) PB is half of PA
    • c) It depends on the radius
    • d) PB is the same length as PA

  3. 3.A tangent from a point P touches a circle with centre O at A. The radius OA is 5 cm and the hypotenuse OP is 13 cm. How long is the tangent PA, in centimetres?

  4. 4.PA and PB are the two tangents drawn from a point P, and PA is 12 cm long. What is the total length of PA and PB, in centimetres?

  5. 5.A tangent PA is 24 cm long and the hypotenuse OP of triangle OAP is 25 cm. How long is the radius OA, in centimetres?

  6. 6.In triangle OAP the angle at A is a right angle and the angle at P is 30°. What is the angle at O, in degrees?

  7. 7.Match each circle and outside point to the length of the tangent from that point, in centimetres.

    • radius 3 cm, point 5 cm from the centre
    • radius 8 cm, point 10 cm from the centre
    • radius 12 cm, point 13 cm from the centre
    • radius 6 cm, point 10 cm from the centre
    • 4
    • 6
    • 5
    • 8

  8. 8.OAPB is a quadrilateral with right angles at A and B, and its angle at P is 70°. What is its angle at O, in degrees?

Answer key — Year 10 Maths — Circles and tangents

  1. 1. 15
  2. 2. d) PB is the same length as PA
  3. 3. 12
  4. 4. 24
  5. 5. 7
  6. 6. 60
  7. 7. radius 3 cm, point 5 cm from the centre → 4; radius 8 cm, point 10 cm from the centre → 6; radius 12 cm, point 13 cm from the centre → 5; radius 6 cm, point 10 cm from the centre → 8
  8. 8. 110