Competition · AMC preparation · step 4 of 4

AMC 8 · 2005 · #7

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

pythagorean-theoremfraction-arithmeticcoordinate-geometry identify-subproblemscoordinate-geometry ↑ Prerequisites: pythagorean-theoremfraction-arithmetic

📏 Short solution 💡 2 insights

Problem

Bill walks

\frac{1}{2}

mile south, then

\frac{3}{4}

mile east, then

\frac{1}{2}

mile south again. How far is his finishing point from his starting point, measured in a straight line?

Pick an answer.

(A)

(B)

$1\tfrac14$

(C)

$1\tfrac12$

(D)

$1\tfrac34$

(E)

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The problem is a path on a flat compass grid, which is the textbook signal for Tool #1 (Draw a Diagram). Sketching the three legs reveals that the two south legs stack into a single vertical segment, and the east leg sits perpendicular to it. Start and finish are then the two non-right corners of a right triangle, so the direct-line distance is the hypotenuse. Once the diagram is drawn, the Pythagorean theorem finishes the job in one line — no algebra needed beyond squaring two fractions.

1STEP 1

Sketch the walk

Sketch it with south as down, east as right: the two south legs share one vertical line with the east step between them.

path: (0,0) → (0,-1/2) → (3/4,-1/2) → (3/4,-1)

2STEP 2

Close the triangle

Draw the start-to-finish segment: the south legs add to 1 and the east leg is $\frac{3}{4}$ , giving a right triangle with that segment as hypotenuse.

south total = 1/2 + 1/2 = 1, east total = 3/4

3STEP 3

Apply the Pythagorean theorem

By the Pythagorean theorem on legs 1 and $\frac{3}{4}$ , the hypotenuse is the square root of $\frac{25}{16}$ — the 3-4-5 triangle scaled by $\frac{1}{4}$ .

d = √(1² + (3/4)²) = √(25/16) = 5/4 = 1 1/4 → (B)

Answer

11/4

The straight-line distance must be less than the total walked,

\frac{1}{2}

\frac{3}{4}

\frac{1}{2}

= 1

\frac{3}{4}

miles, and more than either leg of the triangle on its own, so it has to lie strictly between 1 and 1

\frac{3}{4}

. That rules out (A) 1 and (E) 2, and the diagram makes it clear the answer is only slightly more than 1, which favors (B) 1

\frac{1}{4}

over (C) 1

\frac{1}{2}

. The exact computation confirms (B). The numbers 3-4-5 should also feel familiar: multiplying each by

\frac{1}{4}

gives the triangle

\frac{3}{4}

-1-

\frac{5}{4}

, exactly what came out.

💡Key takeaway

Two south legs separated by an east leg stack into one right triangle — and the start-to-finish line is just the hypotenuse, which here is the familiar 3-4-5 triangle scaled down by $\frac{1}{4}$ .

Sketch the walk
Close the triangle
Apply the Pythagorean theorem

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Sketch the walk

Close the triangle

Apply the Pythagorean theorem

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