NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Irrational numbers

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A walker in Sydney starts at point A and takes two different routes to reach point B. Route 1 is a straight diagonal path across a rectangular field that is 50 metres long and 70 metres wide.

Route 2 goes along the length and then the width of the field. Using Pythagoras’ theorem, what is the exact length of Route 1, in metres?

Give your answer as a simplified exact value.

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Worked solution

  1. The rectangular field has a length of 70 metres and a width of 50 metres. Route 1 is the diagonal path across the field, forming 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 (c) is equal to the sum of the squares of the other two sides (a and b): c² = a² + b².
  3. Substitute the given values: a = 50 m, b = 70 m. So, c² = 50² + 70².
  4. Calculate 50²: 50 × 50 = 2500.
  5. Calculate 70²: 70 × 70 = 4900.
  6. Add the squares: 2500 + 4900 = 7400.
  7. Now find c by taking the square root of 7400: c = sqrt(7400).
  8. Simplify sqrt(7400). Factor 7400: 7400 = 100 × 74 = 100 × 2 × 37. So sqrt(7400) = sqrt(100 × 74) = 10 × sqrt(74).
  9. 74 cannot be simplified further because 74 = 2 × 37, and neither 2 nor 37 is a perfect square.
  10. Therefore, the exact length of Route 1 is 10 × sqrt(74) metres.
  11. Check: 10² × 74 = 100 × 74 = 7400, and 50² + 70² = 2500 + 4900 = 7400. The calculation is consistent.
  12. The exact length of Route 1 is 10*sqrt(74) metres.

Reference answer: 10*sqrt(74) m

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