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# Proofs and Use of the Pythagorean Theorem

## Identify triples and calculate missing sides

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Measuring the World

Before we can put borders around a country or decide where to put a new road, we have to know exactly where those borders are or where that road can be. Land surveyors help gather this information.

#### Why It Matters

Consider the problem of digging to repair a city street. You know that there are electric power lines, telephone lines and gas pipes under the ground, so where do you put your pneumatic drill so that you don't accidentally hit something serious? You will need to find out where the lines and pipes are from a map.

Surveyors create our maps, using a process known as triangulation. Starting from a baseline, they divide an area that they want to map into a series of triangles. Once they have identified the points that mark each triangle, they use very accurate instruments called theodolites to measure the length of each side and the height of the corners. However, the most important tool in triangulation is the Pythagorean theorem.

#### Explore More

A Coast Guard officer spots a ship passing in the distance, and he needs to know how close it is to the lighthouse. He measures the distance to the ship to be 500 yards, and he knows that he is stationed 400 yards from the lighthouse. How far is the ship from the lighthouse? Refer to the figure below.

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