NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Shape-classification algorithms

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Required skills (click to reveal)

A craft designer in Melbourne is creating a pattern using 89 identical regular pentagons and 20 identical isosceles triangles with vertex angles of 36 degrees. The designer uses a flow chart algorithm to classify shapes based on the following steps: (1) Check the number of sides.

If it has 3 sides, classify as a triangle. If it has 4 or more sides, proceed. (2) Check if all sides are equal and all angles are equal.

If yes, classify as a regular polygon. If no, classify as irregular.

What fraction of the total shapes (89 pentagons and 20 triangles) are classified as regular polygons? Answer as a simplified fraction in the form a/b.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The total number of shapes is the sum of 89 regular pentagons and 20 isosceles triangles, so total = 89 + 20 = 109 shapes.
  2. A regular polygon has all sides equal and all angles equal. The pentagons are described as regular, so they meet the criteria for regular polygons.
  3. The 20 triangles are isosceles with vertex angles of 36 degrees. An isosceles triangle with a vertex angle of 36° has base angles of (180° - 36°)/2 = 72°. Since the base angles are not equal to the vertex angle, not all angles are equal, so the triangle is not regular.
  4. The algorithm checks if all sides and angles are equal. For the pentagons: all 5 sides are equal and all 5 angles are equal (each 108°), so they are classified as regular polygons.
  5. For the triangles: they have 3 sides, so the algorithm checks if all sides and angles are equal. The triangle is isosceles, so two sides are equal, but not all three sides are equal unless it is equilateral. Since the vertex angle is 36°, it cannot be equilateral (equilateral triangles have 60° angles), so not all sides are equal. Therefore, the triangle is not regular.
  6. Only the 89 pentagons are classified as regular polygons. The 20 triangles are not regular.
  7. The fraction of shapes classified as regular polygons is 89 / 109.
  8. Check if 89 and 109 have any common factors. 89 is a prime number. 109 is also a prime number. Since both are prime and not equal, the fraction 89/109 is already in simplest form.
  9. Therefore, the simplified fraction is 89/109.
  10. Verify: 89 ÷ 109 ≈ 0.8165, and since 89 and 109 are both prime, no simplification is possible.
  11. The fraction of total shapes classified as regular polygons is 89/109.
  12. Final answer: 89/109.

Reference answer: 89/109

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