NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Required skills (click to reveal)

A craft designer in Brisbane is creating a three-dimensional puzzle piece using digital modelling software. The piece consists of a square base with side length 74 cm, and a right-angled triangular prism is attached to one side.

The triangular cross-section has legs of 74 cm and 86 cm, with the hypotenuse aligned along the base edge. The designer needs to determine the exact length of the internal diagonal that runs from one bottom corner of the square base to the opposite top vertex of the triangular prism's apex.

Using the three-dimensional version of Pythagoras' theorem, what is the exact length of this diagonal in centimetres?

Concept illustrations

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Worked solution

  1. The puzzle piece has a square base with side length 74 cm and a right-angled triangular prism attached to one side. The triangular cross-section has legs of 74 cm and 86 cm, with the hypotenuse aligned along the base edge. The internal diagonal runs from a bottom corner of the base to the opposite top vertex (apex) of the triangular prism.
  2. To find the length of the internal diagonal in 3D space, apply the 3D version of Pythagoras' theorem: d = sqrt(x^2 + y^2 + z^2), where x, y, and z are the perpendicular dimensions of the rectangular box that contains the diagonal.
  3. The diagonal spans three perpendicular directions: the base length (74 cm), the base width (74 cm), and the height of the triangular prism (86 cm). These correspond to the three dimensions of the 3D space.
  4. Substitute the values into the 3D Pythagoras formula: d = sqrt(74^2 + 74^2 + 86^2).
  5. Calculate 74^2: 74 × 74 = 5476.
  6. Calculate 86^2: 86 × 86 = 7396.
  7. Sum the squares: 5476 + 5476 + 7396 = 18,348.
  8. Take the square root of 18,348: d = sqrt(18,348).
  9. Factor 18,348 to simplify: 18,348 = 4 × 4587. Since 4 is a perfect square, sqrt(18,348) = sqrt(4 × 4587) = 2 × sqrt(4587).
  10. Check if 4587 can be simplified further. 4587 ÷ 3 = 1529, and 1529 is not divisible by 3, 5, 7, 11, 13, 17, 19, 23, or 29. No perfect square factors remain, so sqrt(4587) cannot be simplified further.
  11. Thus, the exact length is 2 × sqrt(4587) cm.
  12. The internal diagonal from the bottom corner to the opposite top vertex is exactly 2*sqrt(4587) cm.

Reference answer: 2*sqrt(4587) cm

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