NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Scale, Pythagoras and trigonometry

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A school in Brisbane is ordering custom rectangular desks. Each desk has a length of 63 cm and a width of 43 cm.

Using Pythagoras' theorem, find the exact length of the diagonal across the top of the desk in centimetres.

Rectangle measurementsLength 63 cm; width 43 cm63 cm43 cm
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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.
Choose the angle before naming sides. For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.opp. 3adj. 4hyp. 5θsin θ = 3/5
For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.

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Worked solution

  1. The desk top is a rectangle with length 63 cm and width 43 cm. The diagonal forms a right-angled triangle with these two sides.
  2. Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse (diagonal) equals the sum of the squares of the other two sides.
  3. Let d be the diagonal. Then d² = 63² + 43².
  4. Calculate 63²: 63 × 63 = (60 + 3)² = 60² + 2×60×3 + 3² = 3600 + 360 + 9 = 3969.
  5. Calculate 43²: 43 × 43 = (40 + 3)² = 40² + 2×40×3 + 3² = 1600 + 240 + 9 = 1849.
  6. Add the squares: d² = 3969 + 1849 = 5818.
  7. Take the square root of 5818 to find d: d = sqrt(5818).
  8. Check if 5818 is a perfect square. 76² = 5776, 77² = 5929. Since 5776 < 5818 < 5929, sqrt(5818) is not an integer.
  9. Factor 5818 to simplify: 5818 ÷ 2 = 2909. 2909 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53. 53² = 2809, 59² = 3481, so 2909 is prime. Thus, sqrt(5818) = sqrt(2 × 2909) and cannot be simplified further.
  10. Therefore, d = sqrt(5818) cm. This is the exact length of the diagonal.
  11. Check: 76.2² = (76 + 0.2)² = 76² + 2×76×0.2 + 0.2² = 5776 + 30.4 + 0.04 = 5806.44. 76.3² = 76² + 2×76×0.3 + 0.3² = 5776 + 45.6 + 0.09 = 5821.69. Since 5806.44 < 5818 < 5821.69, sqrt(5818) ≈ 76.27 cm, but the exact value is required.
  12. The exact length of the diagonal is sqrt(5818) cm.

Reference answer: sqrt(5818) cm

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