NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Shape algorithms

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In a community garden in Brisbane, two triangular sections are planned for native plants. Triangle ABC has sides AB = 67 m, BC = 70 m, and AC = 67 m.

Triangle DEF has sides DE = 70 m, EF = 67 m, and DF = 67 m. A student designs an algorithm to test for congruency by comparing corresponding sides and verifying if the matching conditions are met.

Which congruency test applies to confirm the triangles are congruent?

Choose one: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Right angle-Hypotenuse-Side (RHS).

Triangle ABC side measurementsAB = 67 m; AC = 67 m; BC = 70 mABC67 m67 m70 m
Measurements from the question · not to scale.
Triangle DEF side measurementsDE = 70 m; DF = 67 m; EF = 67 mDEF70 m67 m67 m
Measurements from the question · not to scale.
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Classifying from known properties. For a quadrilateral already known to have four right angles, check whether all four sides are equal. Yes gives a square; no gives a rectangle that is not a square. A square remains a rectangle too.4 right anglesRectangle→4 equal sides?Check lengths→Yes: squareNo: non-square
For a quadrilateral already known to have four right angles, check whether all four sides are equal. Yes gives a square; no gives a rectangle that is not a square. A square remains a rectangle too.

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Worked solution

  1. Triangle ABC has sides AB = 67 m, BC = 70 m, and AC = 67 m. Triangle DEF has sides DE = 70 m, EF = 67 m, and DF = 67 m.
  2. List the side lengths of triangle ABC: AB = 67 m, BC = 70 m, AC = 67 m.
  3. List the side lengths of triangle DEF: DE = 70 m, EF = 67 m, DF = 67 m.
  4. Compare corresponding sides: AB = 67 m and EF = 67 m; BC = 70 m and DE = 70 m; AC = 67 m and DF = 67 m.
  5. All three sides of triangle ABC match in length with the three sides of triangle DEF, though not in the same order.
  6. Since all three corresponding sides are equal in length, the triangles are congruent by the Side-Side-Side (SSS) congruency rule.
  7. The SSS rule applies because if three sides of one triangle are equal in length to the three sides of another triangle, the triangles are congruent.
  8. No angles are given, so SAS or ASA cannot be used directly. RHS requires a right angle, which is not stated.
  9. The matching side lengths confirm congruency without needing to check angles or right angles.
  10. Therefore, the correct congruency test is Side-Side-Side (SSS).
  11. The algorithm comparing corresponding sides and verifying matching conditions correctly identifies SSS as the applicable test.
  12. Final verification: all three sides match in length, so SSS is the correct test to confirm congruency.

Reference answer: Side-Side-Side (SSS)

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