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Two friends start walking from the same point in Melbourne. Alice walks 89 metres east and then 80 metres north.
Ben walks 80 metres east and then 89 metres north. What is the exact difference in the straight-line distances they each travel from the starting point?
Simplify your answer using the Pythagorean theorem and algebraic expansion, and give the exact result as a simplified radical expression.
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Worked solution and answer (click to reveal)
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Worked solution
- Alice walks 89 m east and 80 m north, so her displacement from the start forms a right-angled triangle with legs 89 m and 80 m.
- Ben walks 80 m east and 89 m north, so his displacement also forms a right-angled triangle with legs 80 m and 89 m.
- The straight-line distance from the starting point to Alice's final position is given by the Pythagorean theorem: d_A = sqrt(89^2 + 80^2).
- Similarly, Ben's straight-line distance is d_B = sqrt(80^2 + 89^2).
- Since addition is commutative, 89^2 + 80^2 = 80^2 + 89^2, so d_A = d_B.
- Therefore, the difference in their straight-line distances is 0 m.
- Even though the paths differ, the final positions are symmetric with respect to the line y = x, so the distances from the origin are equal.
- No algebraic expansion or simplification is needed because the sum of squares is identical regardless of order.
- The difference in distances is exactly 0 metres.
- Check: 89^2 = 7921, 80^2 = 6400, sum = 14321. sqrt(14321) ≈ 119.67 m for both. Difference is 0.
- The straight-line distances are identical, so the difference is exactly 0 m.
- Final answer: 0 m.
Reference answer: 0 m
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