AMC 10 · 2017 · #4
Grade 8 geometry-2dPick an answer.
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.
Draw it and measure Jerry's route
Setting the side to one makes both routes concrete.
Picking the side to be 1 costs nothing because a percent comparison stays the same at any scale.
6.EE.B.6Draw A DiagramSilvia's route is the diagonal
The straight route is a diagonal.
A square's diagonal closes off a right triangle, so its length is fixed by the two equal legs.
A square's diagonal closes off a right triangle, so its length is fixed by the two equal sides.
▸ Why?
In that right triangle the square on the diagonal equals the two squares on the sides added.
▸ Why?
With both legs equal the triangle has a fixed shape, so the diagonal is always root two times a side.
Find how much shorter the diagonal is
The gap is about 0.59.
An irrational length like √(2) is easiest to compare once you swap in a close decimal.
8.NS.A.2Analyze The UnitsTurn the gap into a percent
Dividing by the longer route gives 30 percent, choice (A).
'Percent shorter' is the saved distance divided by the original distance, not by the new one.
6.RP.A.3Analyze The UnitsCutting 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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