AMC 10 · 2017 · #7

Grade 8 geometry-2d
Archetype Path Length Comparison & Shortest Path └ Sub-archetype Shortest Route in Weighted Network
pythagorean-theorempercentageratio-proportion convert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
On a square field, one person walks along two sides (east then north) to go from the southwest corner to the northeast corner, while the other walks the straight diagonal between the same two corners. Estimate, to the nearest given choice, what percent shorter the diagonal route is than the two-sides route.

Pick an answer.

(A)
$30\%$
(B)
$40\%$
(C)
$50\%$
(D)
$60\%$
(E)
$70\%$

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

This is a paths-on-a-shape problem, so Tool #1 (Draw a Diagram) is the natural start: sketch the square and mark both routes. The two routes split into two independent measurements, so Tool #7 (Identify Subproblems) lets us find Jerry's total and Silvia's total separately. Jerry's total is two sides; Silvia's is the diagonal, which the right triangle inside the square turns into a Pythagorean length. Finally Tool #8 (Analyze the Units) keeps the comparison honest: 'how much shorter' means the gap measured as a percent of Jerry's trip.

1STEP 1

Draw it and measure Jerry's route

Only the ratio matters, so let the side be 1; Jerry walks two sides, so his trip is 1+1.

J = 1 + 1 = 2
2STEP 2

Silvia's route is the diagonal

Silvia's straight line is the square's diagonal—the hypotenuse of a right triangle with legs 1 and 1—so it is √(2).

S = √(1² + 1²) = √(2)
3STEP 3

Find how much shorter the diagonal is

Swap in √(2)≈1.41; the diagonal falls short of Jerry's 2 by the gap 0.59.

J - S = 2 - √(2) ≈ 2 - 1.41 = 0.59
4STEP 4

Turn the gap into a percent

Divide the gap by Jerry's total: 0.59/2≈0.29, i.e. 29%—closest to the listed 30%, choice (A).

(J - S)/J = 0.59/2 ≈ 0.29 = 29% → (A)
Answer
30%
The diagonal of a square is always shorter than going around two sides, so the answer must be a positive percent under 100%. The exact value is (2-√2)/2=1-√2/2≈ 0.293, which rounds to 29% — nearer to 30% than to 40%, confirming (A). Picking a different side length, say 10, gives Jerry 20 and Silvia √(200)≈ 14.14, again about 29% shorter, showing the scale did not matter.
💡Key takeaway

Cutting straight across a square instead of walking two sides saves you about 30% of the distance, because the diagonal is √2 times one side while two sides total 2.

  • Draw it and measure Jerry's route
  • Silvia's route is the diagonal
  • Find how much shorter the diagonal is
  • Turn the gap into a percent

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