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Math 9

Math 9 · Worksheets

Circle geometry

Ten questions of mixed difficulty, covering Circle geometry. Print it, or work through it on screen — the answer key starts on its own page.

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Circle geometry

Math 9 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. A round window has a diameter of 100 cm. A straight bar across it, forming a chord, is 80 cm long. How far is the bar from the centre of the window?

    1. a) 10 cm
    2. b) 60 cm
    3. c) 30 cm
    4. d) 64 cm

  2. AB is a diameter of a circle with centre O, and C is a point on the circle. Angle CAB = 28°. What is angle COB?

    1. a) 14°
    2. b) 124°
    3. c) 62°
    4. d) 56°

  3. A tangent from point P touches a circle with centre O at T. The radius is 5 cm and OP = 13 cm. To the nearest tenth, how long is PT?

    1. a) 18.0 cm
    2. b) 13.9 cm
    3. c) 12.0 cm
    4. d) 8.0 cm

  4. Line PT is tangent to a circle with centre O at point T. In triangle OTP, angle TOP = 52°. What is angle OPT?

    1. a) 90°
    2. b) 128°
    3. c) 52°
    4. d) 38°

  5. Seats in a theatre are on an arc of a circle that also passes through both ends of the stage. From one seat, the stage takes up an angle of 35°. What angle does it take up from another seat on the same arc?

    1. a) 35°
    2. b) 145°
    3. c) 17.5°
    4. d) 70°

  6. AB is a diameter of a circle, and C is another point on the circle. What is angle ACB?

    1. a) 180°
    2. b) 60°
    3. c) 90°
    4. d) 45°

  7. A round drainage pipe has an inside diameter of 100 cm. The surface of the water in it is 80 cm wide, and the pipe is less than half full. How deep is the water?

    1. a) 20 cm
    2. b) 30 cm
    3. c) 10 cm
    4. d) 80 cm

  8. A line from the centre of a circle that is perpendicular to a chord

    1. a) divides the chord in a 1 : 2 ratio
    2. b) bisects the chord
    3. c) is always shorter than the chord
    4. d) is always as long as the radius

  9. A circle has a radius of 10 cm. A chord is 6 cm from the centre. To the nearest tenth, how long is the chord?

    1. a) 8.0 cm
    2. b) 11.7 cm
    3. c) 16.0 cm
    4. d) 23.3 cm

  10. A tangent to a circle meets the radius drawn to the point of tangency at an angle of

    1. a) 45°
    2. b) 180°
    3. c) 60°
    4. d) 90°

Answer key · Circle geometry

Math 9 · maddyhelps.com

  1. c) 30 cm — Use the radius and half the chord: the radius is 50 cm and half the bar is 40 cm, so the distance is √(50² − 40²) = 30 cm. Using the whole diameter and the whole chord gives 60 cm, twice too big, because the right triangle is built on the halves.
  2. d) 56° — Angle CAB is an inscribed angle standing on arc CB, and angle COB is the central angle on the same arc, so it is twice as big: 2 × 28° = 56°. Halving instead of doubling gives 14°, and 62° is angle ABC, which stands on a different arc.
  3. c) 12.0 cm — OT is perpendicular to PT, so triangle OTP has its right angle at T and OP is the hypotenuse: PT² = 13² − 5² = 144, and PT = 12 cm. Adding the squares instead, 169 + 25, treats PT as the hypotenuse and gives 13.9 cm — but PT cannot be longer than OP.
  4. d) 38° — Angle OTP = 90° because the tangent is perpendicular to the radius OT. The angles of triangle OTP add to 180°, so angle OPT = 180° − 90° − 52° = 38°; forgetting the right angle and doing 180° − 52° gives 128°.
  5. a) 35° — Both angles are inscribed angles standing on the same arc — the stage — so they are equal: 35°. Doubling gives 70°, which is the central angle at the circle's centre, not the angle from another seat.
  6. c) 90° — The central angle on a diameter is a straight angle, 180°, and the inscribed angle ACB is half of it: 90°. Any triangle drawn in a semicircle, with the diameter as one side, has a right angle on the circle.
  7. a) 20 cm — The water surface is a chord: with half of it 40 cm and the radius 50 cm, it sits √(50² − 40²) = 30 cm below the centre. The pipe is less than half full, so the depth is 50 − 30 = 20 cm. 30 cm is the distance from the centre, not the depth, and 80 cm is the depth of a pipe more than half full.
  8. b) bisects the chord — The perpendicular from the centre always meets a chord at its midpoint, cutting it into two equal halves. With a radius drawn to one end of the chord, it makes a right triangle, which is how most chord questions are solved.
  9. c) 16.0 cm — The perpendicular from the centre bisects the chord, making a right triangle with hypotenuse 10 (a radius) and one leg 6, so half the chord is √(100 − 36) = 8 cm. The whole chord is 16 cm; stopping at 8 cm is the usual slip.
  10. d) 90° — A tangent touches the circle at one point and is always perpendicular to the radius there. That right angle is what lets you use Pythagoras or the angle sum of a triangle in tangent questions.