2025 AMC 8 Problems/Problem 5
Problem
Betty drives a truck to deliver packages in a neighborhood whose street map is shown below.
Betty starts at the factory (labled 
) and drives to location 
, then 
, then 
, before returning to . What is the shortest distance, in blocks, she can drive to complete the route?
![[asy] unitsize(20); add(grid(8,6)); path w = circle((0,0),0.4); fill(w, white); draw(w); label("$B$",(0,0)); fill(shift((2,4)) * w, white); draw(shift((2,4)) * w); label("$C$",(2,4)); fill(shift((7,3)) * w, white); draw(shift((7,3)) * w); label("$A$",(7,3)); fill(shift((6,5)) * w, white); draw(shift((6,5)) * w); label("$F$",(6,5)); draw((6,-0.2)--(7,-0.2), EndArrow(3)); draw((7,-0.2)--(6,-0.2), EndArrow(3)); draw(shift(6.5, -0.48) * scale(0.03) * texpath("1 block")); draw((8.2,1)--(8.2,2), EndArrow(3)); draw((8.2,2)--(8.2,1), EndArrow(3)); draw(shift(8.88, 1.5) * scale(0.03) * texpath("1 block")); [/asy]](http://latex.artofproblemsolving.com/0/2/f/02f4d70dd8e934b785c58b9e171db52b9bca463e.png)
Solution 1
Each shortest possible path from to follows the edges of the rectangle. The following path outlines a path of 
units:
![[asy] unitsize(20); add(grid(8,6)); draw((6,5)--(7,5)--(7,0)--(0,0)--(0,4)--(2,4)--(2,5)--cycle,green); path w = circle((0,0),0.4); fill(w, white); draw(w); label("$B$",(0,0)); fill(shift((2,4)) * w, white); draw(shift((2,4)) * w); label("$C$",(2,4)); fill(shift((7,3)) * w, white); draw(shift((7,3)) * w); label("$A$",(7,3)); fill(shift((6,5)) * w, white); draw(shift((6,5)) * w); label("$F$",(6,5)); [/asy]](http://latex.artofproblemsolving.com/8/a/d/8ad8c099dd8bd66bb4ff7b18df7d0eee48fccb36.png)
~ zhenghua
Solution 2
Since it's a square grid, you can find the shortest distance using a line diagonally from one point to the other, creating a sort of slope, then find the rise and run of the slope, which also happens to be the shortest distance, repeat this process until you go back to Point , and you should get this:
, which is equal to 
, so your answer will be 
.
~Imhappy62789
Video Solution by Cool Math Problmes
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=2TfegPRg-_k1DEcz&t=257 ~hsnacademy
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination
