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

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A rectangular library shelf in Adelaide has a height of 42 cm and a diagonal length of 65 cm. Using Pythagoras’ theorem, what is the exact length of the base of the shelf in centimetres?

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

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Worked solution and answer (click to reveal)

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Worked solution

  1. The shelf is rectangular, so it forms a right-angled triangle with the height, base, and diagonal as its sides.
  2. Given: height = 42 cm, diagonal = 65 cm. The diagonal is the hypotenuse of the right-angled triangle.
  3. Let the base be b cm. By Pythagoras’ theorem: (base)^2 + (height)^2 = (diagonal)^2.
  4. Substitute known values: b^2 + 42^2 = 65^2.
  5. Calculate 42^2: 42 × 42 = 1764.
  6. Calculate 65^2: 65 × 65 = 4225.
  7. Substitute into equation: b^2 + 1764 = 4225.
  8. Solve for b^2: b^2 = 4225 - 1764 = 2461.
  9. Take the square root of 2461 to find b: b = sqrt(2461).
  10. Check if 2461 is a perfect square: 49^2 = 2401, 50^2 = 2500. 2461 is between them and not a perfect square.
  11. Factor 2461 to simplify: 2461 ÷ 11 = 223.727… not integer. 2461 ÷ 23 = 107, and 23 × 107 = 2461. 23 and 107 are both prime. So sqrt(2461) cannot be simplified further.
  12. The base length is exactly sqrt(2461) cm. This is the exact value.

Reference answer: sqrt(2461)

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