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Math 9

Math 9 · Worksheets

Mixed review · Similarity, circles and data

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

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Mixed review · Similarity, circles and data

Math 9 · maddyhelps.com

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

  1. Triangle ABC is similar to triangle DEF, with AB = 6 cm, BC = 8 cm and DE = 9 cm. How long is EF?

    1. a) 5.3 cm
    2. b) 13.5 cm
    3. c) 11 cm
    4. d) 12 cm

  2. Two tangents from point P touch a circle with centre O at A and B. Angle APB = 50°. What is angle AOB?

    1. a) 50°
    2. b) 130°
    3. c) 40°
    4. d) 100°

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

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

  5. A drawing of a phone is made with a scale factor of 3. The real phone is 14 cm long. How long is the phone in the drawing?

    1. a) 11 cm
    2. b) 4.7 cm
    3. c) 42 cm
    4. d) 17 cm

  6. 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) 70°
    4. d) 17.5°

  7. 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) 8.0 cm
    2. b) 18.0 cm
    3. c) 13.9 cm
    4. d) 12.0 cm

  8. A survey about school supplies asks students to write their name and their family's income. The biggest issue with this survey is

    1. a) privacy
    2. b) use of language
    3. c) cost
    4. d) timing

  9. A 1.5 m tall student casts a 2 m shadow. At the same time, a flagpole casts a 12 m shadow. How tall is the flagpole?

    1. a) 13.5 m
    2. b) 11.5 m
    3. c) 9 m
    4. d) 16 m

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

Answer key · Mixed review · Similarity, circles and data

Math 9 · maddyhelps.com

  1. d) 12 cm — Corresponding sides are in the same ratio: DE/AB = 9/6 = 1.5, so EF = 1.5 × 8 = 12 cm. DE is 3 cm longer than AB, but adding 3 to 8 gives 11 — similar figures multiply, they do not add.
  2. b) 130° — OA ⟂ PA and OB ⟂ PB, so quadrilateral OAPB has two right angles. Its angles add to 360°, so angle AOB = 360° − 90° − 90° − 50° = 130°. Assuming the angle at O equals the angle at P gives 50°.
  3. b) 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°.
  4. b) 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.
  5. c) 42 cm — A scale factor of 3 means every length in the drawing is 3 times the real length: 3 × 14 = 42 cm. Dividing gives 4.7 cm, which would be a reduction, and adding 3 gives 17 cm — scale factors multiply.
  6. 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.
  7. d) 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.
  8. a) privacy — Family income is personal information, and a name on the form means the answers are not anonymous. Students may skip the question or not answer honestly — and income has nothing to do with school supplies anyway.
  9. c) 9 m — The sun makes similar triangles, so height ÷ shadow is the same for both: 1.5/2 = h/12, giving h = 0.75 × 12 = 9 m. Flipping one ratio gives 16 m; a quick check catches it, since the student is shorter than their shadow, so the pole must be shorter than its shadow too.
  10. d) 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.