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Math 9

Math 9 · Worksheets

Similarity and transformations

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

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Similarity and transformations

Math 9 · maddyhelps.com

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

  1. On a map with a scale of 1 : 5 000 000, two towns are 5 cm apart. What is the real distance between them?

    1. a) 2500 km
    2. b) 25 km
    3. c) 25 000 km
    4. d) 250 km

  2. Two polygons are similar exactly when

    1. a) corresponding sides are proportional, even if the angles are not equal
    2. b) corresponding angles are equal and corresponding sides are proportional
    3. c) they have the same number of sides and the same shape name
    4. d) corresponding angles are equal, even if the sides are not proportional

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

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

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

  5. A regular pentagon has rotation symmetry of order 5. What is its smallest angle of rotation symmetry?

    1. a) 36°
    2. b) 108°
    3. c) 72°
    4. d) 60°

  6. 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) 17 cm
    4. d) 42 cm

  7. The point (3, 2) is on a shape that has the y-axis as a line of symmetry. Which other point must be on the shape?

    1. a) (2, 3)
    2. b) (−3, −2)
    3. c) (3, −2)
    4. d) (−3, 2)

  8. In triangle ABC, D is on AB and E is on AC, with DE parallel to BC, so triangle ADE is similar to triangle ABC. AD = 4 cm, DB = 6 cm and DE = 5 cm. How long is BC?

    1. a) 11 cm
    2. b) 7.5 cm
    3. c) 2 cm
    4. d) 12.5 cm

  9. How many lines of symmetry does a regular hexagon have?

    1. a) 6
    2. b) 3
    3. c) 12
    4. d) 2

  10. Which statement about a rectangle that is not a square is true?

    1. a) It has 2 lines of symmetry and rotation symmetry of order 4.
    2. b) It has 4 lines of symmetry and rotation symmetry of order 4.
    3. c) It has 4 lines of symmetry and rotation symmetry of order 2.
    4. d) It has 2 lines of symmetry and rotation symmetry of order 2.

Answer key · Similarity and transformations

Math 9 · maddyhelps.com

  1. d) 250 km — Multiply first: 5 × 5 000 000 = 25 000 000 cm. There are 100 000 cm in a kilometre (100 cm in a metre, 1000 m in a kilometre), so that is 250 km. Most wrong answers here come from the unit conversion, not the scale.
  2. b) corresponding angles are equal and corresponding sides are proportional — Polygons need both conditions. A square and a long rectangle have equal angles but are not similar, and a square and a rhombus have proportional sides but are not similar; triangles are the special case where equal angles are enough.
  3. d) 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.
  4. a) 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.
  5. c) 72° — A full turn of 360° is shared equally among the 5 positions where the pentagon looks the same: 360° ÷ 5 = 72°. 108° is each interior angle of the pentagon, a different angle that is easy to mix up with this one.
  6. d) 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.
  7. d) (−3, 2) — Reflecting in the y-axis swaps left and right, so the x-coordinate changes sign and the y-coordinate stays: (−3, 2). (3, −2) is the reflection in the x-axis, and (−3, −2) is a half-turn about the origin.
  8. d) 12.5 cm — In the big triangle, the sides that correspond to AD and DE are AB and BC, and AB = 4 + 6 = 10 cm. So BC/5 = 10/4 and BC = 12.5 cm. Using DB = 6 instead of AB = 10 gives 7.5 cm; DB is only part of a side.
  9. a) 6 — There are 3 lines through opposite corners and 3 more through the midpoints of opposite sides, 6 in all. Counting only the lines through corners gives 3. A regular polygon has as many lines of symmetry as it has sides.
  10. d) It has 2 lines of symmetry and rotation symmetry of order 2. — Only the two lines through the midpoints of opposite sides are lines of symmetry; a diagonal is not, because folding along it does not match the corners, and that is the usual mistake. A half-turn matches the rectangle to itself but a quarter-turn does not, so the order of rotation is 2.