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

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Required skills (click to reveal)

In a community garden in Melbourne, a rectangular plot is divided into two congruent right-angled triangular sections by a diagonal path. The length of the shorter side of the rectangle is 33 metres and the longer side is 83 metres.

A gardener walks along the diagonal path from one corner to the opposite corner. Using geometric reasoning and the properties of congruent triangles, determine the exact length of the diagonal path in metres.

Provide only the exact value as a simplified radical expression.

Concept illustrations

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A counterexample to a universal claim. A 3 cm by 2 cm rectangle has four right angles but unequal adjacent sides. It disproves the claim that every rectangle is a square. One counterexample is enough.3 cm2 cmRectangle ≠ necessarily square
A 3 cm by 2 cm rectangle has four right angles but unequal adjacent sides. It disproves the claim that every rectangle is a square. One counterexample is enough.
Adjacent angles on a straight line. The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.65°115°65° + 115° = 180°
The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The rectangular plot has sides of 33 m and 83 m. The diagonal path divides the rectangle into two congruent right-angled triangles.
  2. In a right-angled triangle, the diagonal (hypotenuse) can be found using Pythagoras' theorem: c² = a² + b², where a and b are the legs and c is the hypotenuse.
  3. Substitute the given side lengths: a = 33 m, b = 83 m.
  4. Calculate a²: 33² = 33 × 33 = 1089.
  5. Calculate b²: 83² = 83 × 83 = 6889.
  6. Add the squares: 1089 + 6889 = 7978.
  7. Find the square root of 7978 to get the diagonal length: c = sqrt(7978).
  8. Factor 7978 to simplify the radical: 7978 ÷ 2 = 3989. 3989 is a prime number (verified by testing divisibility up to sqrt(3989) ≈ 63.2).
  9. So, 7978 = 2 × 3989. Since 3989 is prime and not a perfect square, the radical cannot be simplified further.
  10. Therefore, the diagonal length is sqrt(7978) metres.
  11. The diagonal path length is exactly sqrt(7978) m, as no further simplification is possible.
  12. Check: 89² = 7921, 90² = 8100. 7978 is between them, and since 7978 = 2 × 3989 and 3989 is prime, the radical is in simplest form.

Reference answer: sqrt(7978)

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