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

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A craft designer in Brisbane is creating a series of triangular wall art pieces based on a standard isosceles triangle with side lengths 58 cm, 58 cm, and 87 cm. An algorithm is used to determine if a new design is congruent to the standard using the SSS (side-side-side) congruence criterion.

A new design has side lengths 58 cm, 58 cm, and 87 cm. What is the exact result of the algorithm's comparison?

Answer exactly as 'congruent' or 'not congruent'.

Concept illustrations

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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.
Squares belong to two broader families. A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.RectangleRhombusSquare
A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.

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Explanation

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Worked solution

  1. The standard triangle has side lengths 58 cm, 58 cm, and 87 cm. This is an isosceles triangle with two equal sides.
  2. The new design has side lengths 58 cm, 58 cm, and 87 cm. These match the side lengths of the standard triangle exactly.
  3. The SSS (Side-Side-Side) congruence criterion states that if all three sides of one triangle are equal in length to the corresponding sides of another triangle, then the triangles are congruent.
  4. The algorithm compares the three side lengths of the new design to those of the standard triangle.
  5. The first side: 58 cm matches 58 cm.
  6. The second side: 58 cm matches 58 cm.
  7. The third side: 87 cm matches 87 cm.
  8. All three sides are equal in length to the corresponding sides of the standard triangle.
  9. By the SSS congruence criterion, the two triangles are congruent.
  10. The algorithm returns 'congruent' because all three sides match exactly.
  11. The result is confirmed: the new design is identical in side lengths to the standard triangle.
  12. Therefore, the algorithm's comparison result is exactly 'congruent'.

Reference answer: congruent

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