NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Real numbers

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A walker in Brisbane starts at point A and walks 94 metres north, then turns 90 degrees west and walks 11 metres to reach point B. Using Pythagoras' theorem, calculate the exact straight-line distance from point A to point B in metres.

Express your answer as a simplified exact value.

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Rational numbers inside the real numbers. Rational and irrational numbers are disjoint parts of the real numbers. The examples 4 and 1/2 are rational; √2 and π are irrational.Real numbersRationalIrrational4 1/2√2 π
Rational and irrational numbers are disjoint parts of the real numbers. The examples 4 and 1/2 are rational; √2 and π are irrational.

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Worked solution

  1. The walker moves 94 metres north and then 11 metres west, forming a right-angled triangle with point A as the starting point and point B as the destination.
  2. The straight-line distance from A to B is the hypotenuse of this right-angled triangle, with legs of 94 m and 11 m.
  3. Pythagoras' 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): c² = a² + b².
  4. Substitute the given values: a = 94, b = 11. So c² = 94² + 11².
  5. Calculate 94²: 94 × 94 = (90 + 4)² = 90² + 2×90×4 + 4² = 8100 + 720 + 16 = 8836.
  6. Calculate 11²: 11 × 11 = 121.
  7. Add the squares: c² = 8836 + 121 = 8957.
  8. Take the square root of 8957 to find c: c = sqrt(8957).
  9. Check if 8957 can be simplified: factor 8957. It is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, or 97. No perfect square factors exist.
  10. Since 8957 has no square factors, sqrt(8957) is already in simplest radical form.
  11. The exact straight-line distance from point A to point B is sqrt(8957) metres.
  12. Verification: 94² = 8836, 11² = 121, sum = 8957. No perfect square factor divides 8957, so the radical cannot be simplified further.

Reference answer: sqrt(8957)

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