NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Pythagorean theorem

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In a community garden in Melbourne, a rectangular vegetable plot has one side measuring 41 metres and the diagonal measuring 32√5 metres. Using Pythagoras’ theorem, what is the exact length, in metres, of the other side of the plot?

Provide your answer as a single exact value using a radical or integer.

Concept illustrations

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A right triangle. The marked angle is 90°. 3² + 4² = 9 + 16 = 25 = 5². All sides use the same unit.345
The marked angle is 90°. 3² + 4² = 9 + 16 = 25 = 5². All sides use the same unit.
A square with area forty-nine. 49 unit squares form a 7 by 7 square. √49 = 7.7 units across7
49 unit squares form a 7 by 7 square. √49 = 7.

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Worked solution and answer (click to reveal)

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Worked solution

  1. The vegetable plot is rectangular, so it forms a right-angled triangle with the diagonal as the hypotenuse. One side is given as 41 m, and the diagonal (hypotenuse) is 32√5 m. Let the unknown side be x metres.
  2. Apply Pythagoras’ theorem: a² + b² = c², where a and b are the legs of the right triangle, and c is the hypotenuse.
  3. Substitute the known values: 41² + x² = (32√5)².
  4. Calculate 41²: 41 × 41 = 1681.
  5. Calculate (32√5)²: 32² × (√5)² = 1024 × 5 = 5120.
  6. Substitute into the equation: 1681 + x² = 5120.
  7. Solve for x²: x² = 5120 - 1681 = 3439.
  8. Take the square root of both sides: x = √3439.
  9. Check if 3439 can be simplified: factor 3439. Try dividing by small primes: 3439 ÷ 19 = 181 (since 19 × 181 = 3439). 19 and 181 are both prime, so √3439 cannot be simplified further.
  10. Therefore, the exact length of the other side is √3439 metres.
  11. Verify: 41² + (√3439)² = 1681 + 3439 = 5120, and (32√5)² = 1024 × 5 = 5120. The values match.
  12. The other side of the plot is exactly √3439 metres.

Reference answer: sqrt(3439)

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