Practice Maths

Topic Review — Similarity and Scale

Mixed Practice — L37 – L39

This review covers all lessons in the Similarity and Scale topic. Try each question before checking your answers.

Review Questions

  1. Similar figures — identifying and scale factors. Fluency

    1. Two similar triangles have corresponding sides 8 cm and 12 cm. Find the scale factor from smaller to larger.
    2. ▵ABC ~ ▵DEF with AB = 10, DE = 15, BC = 14. Find EF.
    3. Two similar pentagons have a scale factor of 2. If the smaller has perimeter 30 cm, find the perimeter of the larger.
    4. Two similar figures have areas 36 cm² and 144 cm². Find the scale factor of their sides.
    5. State the similarity test (AA, SAS, or SSS) for each pair:
      1. ▵PQR with ∠P = 45°, ∠Q = 65° and ▵STU with ∠S = 45°, ∠T = 65°.
      2. ▵LMN with sides 4, 6, 8 and ▵XYZ with sides 6, 9, 12.

  2. Finding unknown sides. Fluency

    1. In the figure, DE ∥ BC, AD = 6, DB = 4, DE = 9. Find BC.
    2. Two similar rectangles: the first is 5 cm × 8 cm, the second has its shorter side = 7.5 cm. Find the longer side.
    3. A 2 m stick casts a shadow of 1.6 m. At the same time, a building casts a shadow of 20 m. Find the height of the building.
    4. ▵RST ~ ▵UVW. RS = 7, ST = 11, TR = 9, UV = 21. Find VW and WU.

  3. Scale drawings and maps. Understanding

    1. A map has scale 1 : 20 000. Two points are 6.5 cm apart on the map. Find the actual distance in metres.
    2. A road is 8.4 km long. Find its length on a map with scale 1 : 200 000 (answer in cm).
    3. A floor plan (scale 1 : 50) shows a room as 6 cm × 4.5 cm. Find the actual area of the room in m².
    4. Find the scale of a drawing where 3 cm represents 45 m. Write as a ratio 1 : n.
    5. A park appears as 8 cm × 5 cm on a map with scale 1 : 5000. Find the actual area in m² and in hectares.

  4. Combined similarity and scale. Problem Solving

    1. A scale model of a bridge is built at 1 : 200. The model is 45 cm long and 6 cm wide.
      1. Find the actual length and width of the bridge.
      2. The actual bridge deck has an area of 1296 m². Find the model deck area in cm², using the area scale factor.
      3. Verify by multiplying the model dimensions.
    2. In a right-angled triangle, the altitude from the right angle to the hypotenuse has length 6 cm. The two segments of the hypotenuse are 4 cm and 9 cm.
      1. Verify that the altitude² = product of the two segments (geometric mean relation).
      2. Show that the two smaller triangles formed are each similar to the original large triangle (name the similarity test).
    3. A photograph 15 cm × 10 cm is to be printed at two different enlargement scales: 150% and 250%. For each enlargement:
      1. Find the new dimensions.
      2. Find the area of each enlargement.
      3. State the area scale factor compared to the original.

  5. Mixed extended response. Problem Solving

    1. A surveyor uses a map (scale 1 : 10 000) to plan a triangular paddock. On the map, the three sides of the paddock measure 4.5 cm, 6.2 cm, and 7.8 cm.
      1. Find the actual side lengths of the paddock.
      2. Calculate the perimeter of the paddock.
      3. Using Heron’s formula (or another method), estimate the area of the paddock, given that it is approximately a right-angled triangle. State the area in hectares.
      4. Fencing costs $28 per metre. Find the total fencing cost.
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