NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Congruence and similarity algorithms

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An algorithm is designed to determine if two triangles are similar using the SSS (side-side-side) similarity criterion. Triangle ABC has side lengths AB = 6 cm, BC = 8 cm, and AC = 10 cm.

Triangle DEF has side lengths DE = 9 cm, EF = 12 cm, and DF = 15 cm. The algorithm checks if the ratios of corresponding sides are equal.

What is the exact ratio of corresponding sides used in the algorithm to confirm similarity? Answer as a simplified fraction in the form a/b.

Triangle ABC side measurementsAB = 6 cm; AC = 10 cm; BC = 8 cmABC6 cm10 cm8 cm
Measurements from the question · not to scale.
Triangle DEF side measurementsDE = 9 cm; DF = 15 cm; EF = 12 cmDEF9 cm15 cm12 cm
Measurements from the question · not to scale.

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Explanation

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Worked solution

  1. Given: Triangle ABC with sides AB = 6 cm, BC = 8 cm, AC = 10 cm. Triangle DEF with sides DE = 9 cm, EF = 12 cm, DF = 15 cm. The algorithm uses the SSS similarity criterion, which requires the ratios of corresponding sides to be equal.
  2. To apply SSS similarity, identify corresponding sides based on side length order. Arrange both triangles' sides in ascending order to match corresponding sides.
  3. Triangle ABC side lengths in ascending order: 6 cm, 8 cm, 10 cm.
  4. Triangle DEF side lengths in ascending order: 9 cm, 12 cm, 15 cm.
  5. Match corresponding sides: 6 cm → 9 cm, 8 cm → 12 cm, 10 cm → 15 cm.
  6. Calculate the ratio of the first pair: 6 / 9 = 2 / 3.
  7. Calculate the ratio of the second pair: 8 / 12 = 2 / 3.
  8. Calculate the ratio of the third pair: 10 / 15 = 2 / 3.
  9. All three ratios are equal to 2/3, confirming that the triangles are similar by SSS similarity.
  10. The ratio used in the algorithm to confirm similarity is 2/3.
  11. Check: 6/9 = 2/3, 8/12 = 2/3, 10/15 = 2/3. All match exactly.
  12. Therefore, the exact ratio of corresponding sides used in the algorithm is 2/3.

Reference answer: 2/3

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