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

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Two triangular frames for a sports equipment rack in Brisbane are to be checked for congruency using an algorithm. Frame A has sides AB = 81 cm, BC = 82 cm, and AC = 115 cm.

Frame B has sides DE = 82 cm, EF = 115 cm, and DF = 81 cm. The algorithm tests for congruency by comparing corresponding sides.

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).

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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. Given: Triangle A has sides AB = 81 cm, BC = 82 cm, AC = 115 cm. Triangle B has sides DE = 82 cm, EF = 115 cm, DF = 81 cm.
  2. List the side lengths of both triangles: Triangle A: 81 cm, 82 cm, 115 cm. Triangle B: 81 cm, 82 cm, 115 cm.
  3. Compare the three side lengths of Triangle A and Triangle B. All three sides are equal in length: 81 cm, 82 cm, 115 cm.
  4. Since all three corresponding sides are equal in length, the triangles are congruent by the Side-Side-Side (SSS) congruency rule.
  5. The SSS rule applies because if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
  6. No angles are given, so SAS or ASA cannot be used. RHS requires a right angle, which is not indicated.
  7. The algorithm compares corresponding sides, and since all three pairs match exactly, SSS is the correct test.
  8. Verify correspondence: AB = 81 cm corresponds to DF = 81 cm; BC = 82 cm corresponds to DE = 82 cm; AC = 115 cm corresponds to EF = 115 cm.
  9. All three pairs of sides are equal, so the triangles are congruent under SSS.
  10. The algorithm correctly identifies congruency using SSS because all three sides match.
  11. Therefore, the correct congruency test is Side-Side-Side (SSS).
  12. Final check: All side lengths match exactly, and no other test is applicable without angle data.

Reference answer: Side-Side-Side (SSS)

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