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2017 AMC 10A Problem 7

2017 AMC 10A Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Triangles: Area & Pythagorean · difficulty 1 of 5 · about 1 min · position P1-10

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

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

Answer

Key insight

With side 1, Jerry walks 2 and Silvia walks the diagonal sqrt(2) of about 1.41, so she saves about 0.59 out of 2, roughly 30%.

Solution

The size of the field does not matter, so let each side be $1$ .

Jerry walks two full sides: $1 + 1 = 2$ .

Silvia walks the diagonal, the hypotenuse of a right isosceles triangle with legs $1$ , so her distance is $\sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414$ .

She saves $2 - 1.414 = 0.586$ , which as a fraction of Jerry's trip is
$\frac{0.586}{2} \approx 0.293 \approx 29\%.$

The closest choice is $\boxed{\textbf{(A)}\ 30\%}$ .

Why this works

"How much shorter" is a percent decrease, so the base is the longer (original) trip. The diagonal of a square is $\sqrt{2}$ times its side, so the ratio diagonal-to-two-sides is $\frac{\sqrt2}{2} \approx 0.707$ , and the saving is $1 - 0.707 \approx 0.29$ regardless of the field's size.

The trap

Computing the saving as a fraction of Silvia's distance (about 41%) instead of Jerry's, or forgetting the diagonal is sqrt(2), not 1.5.

Common mistakes

Computing the saving as a fraction of Silvia's distance (about 41%) instead of Jerry's, or forgetting the diagonal is sqrt(2), not 1.5.
Reporting the ratio $\sqrt2/2 \approx 70\%$ (choice E), which is how long her trip is, not how much shorter.

Techniques

Bound the quantity above/below or estimate to pin it down · Set up the equation/formula and compute; no special trick needed