PROBLEM: Four Dogs

Click **here**
for a statement of the problem.

## Discussion/ Solution?:

Four dogs, **A**, **B**, **C**, and **D**, are located at the
vertices of a square. They start to run at the same time and at the same
speed. **A** runs toward **B**, **B** runs toward **C**, **C**
runs toward **D**, and **D** runs toward **A**.

Eventually they all meet in the center of the square? Why?

The dogs will meet at the center of the square because they each follow
along an arc of a circle that passes through the center of the square and
two vertices of the square.

Each path is actually one-fourth of the circle it follows. Let **2s**
represent the length of the side of the square. The radius of each circle
is **s**. So the circumference of each circle is C = 2pr = 2ps. Thus,
each dog travels a distance of d = (2ps)/4 = ps**/**2 .

Each dog begins his journey at the corners of the square. So each dog
begins at the intersection of a vertex and a diagonal. The path of each
dog doesn't cross the diagonals again until they meet in the middle of the
square.

Click **here **for an animation of the dog chase.

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