AMC 10 · 2025 · #7

Grade 8 geometry-2d

Archetype Path Length Comparison & Shortest Path └ Sub-archetype Sum Known Path Segments

pythagorean-theoremcoordinate-geometrylinear-equations-one-var convert-to-algebra ↑ Prerequisites: pythagorean-theorem

📏 Medium solution 💡 3 insights

Problem

Frances stands 15 m due south of a locked gate on an east-west fence. A box sits on the fence line x m east of the locked gate. One open gate is 9 m east of the box; another open gate is 8 m west of the locked gate. To reach the box Frances walks straight to an open gate, through it, then straight to the box. Both open gates give the same total walking distance. Find x.

Pick an answer.

(A)

$3 \frac{2}{7}$

(B)

$3 \frac{3}{7}$

(C)

$3 \frac{4}{7}$

(D)

$3 \frac{5}{7}$

(E)

$3 \frac{6}{7}$

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The scene is all about positions and paths, so drawing it on a grid is the first move: it reveals that Frances, being straight south of the fence, sees every gate at the top of a right triangle with a 15 m vertical leg. That turns each diagonal walk into a Pythagorean length. Then name the box distance x, write each route's total length, set the two equal, and solve the resulting equation.

1STEP 1

Put the scene on a grid

Locked gate at the origin, east positive: Frances (0,-15), box (x,0), west gate (-8,0), east gate (x+9,0).

F=(0,-15), box=(x,0), W=(-8,0), E=(x+9,0)

2STEP 2

Measure the west route

West route: the diagonal is the 8-15-17 hypotenuse 17, plus x+8 along the fence, so the total is x+25.

√(8²+15²)=17, west total=17+(x+8)=x+25

3STEP 3

Measure the east route

East route: the same right triangle gives the diagonal √((x+9)²+225), then only 9 m back west to the box.

east total=√((x+9)²+225)+9

4STEP 4

Set the routes equal and solve

Equal totals give x+16=√((x+9)²+225); squaring kills x² and leaves 14x=50, so x=25/7=3 4/7, choice (C).

x+16=√((x+9)²+225) → 14x=50 → x=25/7=3 4/7

Answer

3 4/7

Plug x=25/7 back in. West route: x+25=25/7+25=200/7≈ 28.57. East route: x+9=88/7, so the diagonal is √((88/7)²+225)=√(7744/49+11025/49)=√(18769/49)=137/7, and adding 9 gives 137/7+63/7=200/7≈ 28.57. Both routes match, and 3 4/7 sits in the middle of the answer choices, so (C) is consistent.

💡Key takeaway

Drop everything onto a grid, use the right triangles to measure the diagonal walks, set the two equal-length routes equal, and solve for the unknown.

Put the scene on a grid
Measure the west route
Measure the east route
Set the routes equal and solve

Put the scene on a grid

Measure the west route

Measure the east route

Set the routes equal and solve

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