NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Geometric construction algorithms

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Required skills (click to reveal)

A sports equipment manufacturer in Sydney is designing a custom triangular frame for a new training tool. The frame must be constructed using an algorithm that verifies the triangle inequality and applies the Pythagorean theorem to confirm a right angle.

The algorithm uses two legs of length 35 cm and 37 cm. Determine the exact length of the hypotenuse, in centimetres, using the Pythagorean theorem.

Express your answer in simplest radical form.

Concept illustrations

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The problem states that the triangle has two legs of 35 cm and 37 cm, and asks for the hypotenuse using the Pythagorean theorem. This applies because the triangle is right-angled, and the hypotenuse is opposite the right angle.
  2. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b). The formula is c² = a² + b².
  3. Substitute the given side lengths: a = 35 cm, b = 37 cm. So c² = 35² + 37².
  4. Calculate 35²: 35 × 35 = 1225.
  5. Calculate 37²: 37 × 37 = 1369.
  6. Add the squares: 1225 + 1369 = 2594.
  7. Now solve for c: c = √2594.
  8. Factor 2594 to simplify the square root. Check for perfect square factors: 2594 ÷ 2 = 1297. 1297 is a prime number (verified by testing divisibility up to √1297 ≈ 36.0).
  9. So √2594 = √(2 × 1297). Since 1297 is prime and not a perfect square, and 2 is not a square factor, the expression cannot be simplified further.
  10. Therefore, the hypotenuse is exactly √2594 cm.
  11. Check: 35² + 37² = 1225 + 1369 = 2594. √2594 is the only value that satisfies c² = 2594.
  12. The exact length of the hypotenuse is √2594 cm.

Reference answer: sqrt(2594) cm

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