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Year 11 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.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
    • 5
    • 4
    • 6
    • 8

  2. 2.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?

  3. 3.Three straight lines are drawn on this circle. Tap the tangent.

    Write the number of the part.

    123

  4. 4.A tangent touches a circle with centre O at the point P. What is the angle between the tangent and the radius OP?

    • a) 45°
    • b) 180°
    • c) 60°
    • d) 90°

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

  6. 6.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?

  7. 7.Sort each point by how many tangents can be drawn from it to a given circle.

    Groups: No tangents · Exactly one tangent · Exactly two tangents

    • A point inside the circle
    • A point on the circle itself
    • The centre of the circle
    • A point outside the circle

  8. 8.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?

Answer key — Year 11 Maths — Circles and tangents

  1. 1. 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
  2. 2. 24
  3. 3. 1 — The tangent
  4. 4. d) 90°
  5. 5. c) PB is the same length as PA
  6. 6. 7
  7. 7. A point inside the circle → No tangents; The centre of the circle → No tangents; A point on the circle itself → Exactly one tangent; A point outside the circle → Exactly two tangents
  8. 8. 12