NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Algebraic expressions and equations

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Two friends start walking from the same point in Brisbane. Alice walks 74 metres east and then 61 metres north.

Ben walks 61 metres east and then 74 metres north. Using the Pythagorean theorem and algebraic simplification, find the exact difference in the straight-line distances they each travel from the starting point.

Express your answer as a simplified radical expression.

Concept illustrations

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Combining like terms as equal tiles. Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.xxxxx3x2x3x + 2x = 5x
Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.
Distributive law as a rectangle. A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.x433x123(x + 4) = 3x + 12
A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Alice walks 74 m east and 61 m north, so her straight-line distance from the start is the hypotenuse of a right triangle with legs 74 m and 61 m.
  2. Ben walks 61 m east and 74 m north, so his straight-line distance is the hypotenuse of a right triangle with legs 61 m and 74 m.
  3. The Pythagorean theorem states that for a right triangle with legs a and b, and hypotenuse c, c^2 = a^2 + b^2.
  4. For Alice: d_A^2 = 74^2 + 61^2 = 5476 + 3721 = 9197.
  5. For Ben: d_B^2 = 61^2 + 74^2 = 3721 + 5476 = 9197.
  6. Both d_A^2 and d_B^2 equal 9197, so d_A = sqrt(9197) and d_B = sqrt(9197).
  7. The difference in straight-line distances is d_A - d_B = sqrt(9197) - sqrt(9197) = 0.
  8. Since both paths result in the same hypotenuse length, the difference is exactly zero.
  9. No simplification of a radical is needed because the difference is zero.
  10. The algebraic expression for the difference is sqrt(74^2 + 61^2) - sqrt(61^2 + 74^2), which simplifies to 0.
  11. The symmetry in the legs (74 and 61) means the hypotenuse is identical regardless of order.
  12. Therefore, the exact difference in straight-line distances is 0 metres.

Reference answer: 0 m

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