From a train station, one train heads north, and another heads east. Some time later, the northbound train has traveled 12 miles, and the eastbound train has traveled 16 miles. How far apart are the two trains, measured in a straight line?

9514 1404 393

  20 miles

Step-by-step explanation:

The north and east distances form the legs of a right triangle. The straight-line distance between the trains is the hypotenuse of that triangle. The Pythagorean theorem can be used to find the length of the hypotenuse (d).

  d² = 12² +16²

  d² = 144 +256 = 400

  d = √400 = 20

The two trains are 20 miles apart.

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Additional comment

You may recognize the given distances are in the ratio 3 : 4.  You may recall that side ratios of 3 : 4 : 5 make a right triangle. If so, you recognize that the straight-line distance is (4 miles)×5 = 20 miles. (3:4:5 right triangles show up often in algebra and geometry problems, so might be something you want to look for.)

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