NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Scale, Pythagoras and trigonometry

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In a community garden in Melbourne, a rectangular vegetable plot measures 87 m in length and 39 m in width. A diagonal pathway is planned from one corner to the opposite corner.

Using Pythagoras' theorem, find the exact length of the diagonal pathway in metres.

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Worked solution

  1. The vegetable plot is rectangular with length 87 m and width 39 m. The diagonal pathway forms a right-angled triangle with the length and width as the two perpendicular sides.
  2. Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse (diagonal) equals the sum of the squares of the other two sides.
  3. Let d be the diagonal length. Then, d² = 87² + 39².
  4. Calculate 87²: 87 × 87 = (80 + 7)² = 80² + 2×80×7 + 7² = 6400 + 1120 + 49 = 7569.
  5. Calculate 39²: 39 × 39 = (40 - 1)² = 40² - 2×40×1 + 1² = 1600 - 80 + 1 = 1521.
  6. Add the squares: d² = 7569 + 1521 = 9090.
  7. Take the square root of 9090 to find d: d = sqrt(9090).
  8. Factor 9090 to simplify: 9090 = 9 × 1010 = 9 × 10 × 101 = 9 × 2 × 5 × 101. So sqrt(9090) = sqrt(9 × 1010) = 3 × sqrt(1010).
  9. Check if 1010 can be simplified further: 1010 = 2 × 5 × 101. No square factors other than 1, so sqrt(1010) cannot be simplified.
  10. Therefore, d = 3 × sqrt(1010) metres. This is the exact length.
  11. Verify by squaring 3×sqrt(1010): (3×sqrt(1010))² = 9 × 1010 = 9090, which matches d².
  12. The diagonal pathway is exactly 3×sqrt(1010) metres long.

Reference answer: 3*sqrt(1010) m

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