SOLUTION: Joe start walking due north, Bob starts waling due east. When they stop Bob has walked 1 mile further than Joe and they are 5 miles apart as the crow flies. How far did Bob walk?

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Let's consider this problem geometrically. We'll call Joe's starting point J, Bob's starting point B and the point at which they meet we'll call M. The angle JMB is a right angle because Joe is walking north and Bob is walking east. We also know that the line from Joe's starting point to Bob's starting point is 5 miles. This line forms the hypotenuse of the right triangle JMB. We also know that the distance JM is one mile more than the distance MB.

Now let's collect these facts:

1. The triangle with verticies JMB is a right triangle


2. JB is the hypotenuse of the triangle and is 5 miles long


3. JM = MB + 1

For simplicity, let's call JM x and JB y. From the Pythagorean theorem, we know that , this problem tells us that h = 5 and . This allow's us to restate the formula for a right triamgle as:



completing the square, we get

adding like terms we get

subtrating 25 from both sides of the equation from we get .
Notice that we can factor 2 our of the equation so .
This equation factors to so y = -4 or y = 3. A distance of -4 does not make sense, so we set y to 3 and this tells us that . Remember y is the distance of the line MB which is the distance Bob walked and x is the distance of the line JM, which is the distance Joe walked.