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Latitude and Longitude:
A Real Life Example of the Pythagorean Theorem

Robin A. Ward,California Polytechnic State University-San Luis Obispo

 Audience: Grades 7-8, or higher

 Mathematical Topics: Pythagorean Theorem, coordinate graphing


latitude and longitude coordinates of NASA sites
map of U.S.

The Activity:

Some students may have noticed that the shortest distance between any two locations is NOT found by moving along the latitude lines and then along the longitude lines (shown in purple). Instead, the shortest distance between two locations is found by drawing a straight line connecting the two locations (shown in red ).

Ask students the following question: "Using the lines of latitude and longitude, what is the distance (in degrees) if you were to travel from NASA Dryden to NASA Ames?"

Based on the map, NASA Dryden is located (approximately) at 117 degrees longitude and 34 degrees latitude. NASA Ames is located (approximately) at 122 degrees longitude and 37 degrees latitude. Using this information, students can use the Pythagorean Theorem to find the distance (measured in degrees) between these two locations.

Students will first compute the vertical and horizontal distances between the two locations using the longitude and latitude readings, respectively. The change in longitude between the two locations is 5 degrees, since we are moving from 117 degrees longitude to 122 degrees longitude. The change in latitude between the two locations is 3 degrees, since we are moving from 37 degrees to 34 degrees as we travel from NASA Dryden to NASA Ames. These two distances form the legs of a right triangle.

Knowing the length of the legs of the triangle, students can now compute the length of the hypotenuse (which represents the shortest distance between the two locations) of the triangle by using the Pythagorean Theorem.

Promote a discussion with students as to whether the solution of 5.83 degrees makes sense. Recall that given any right triangle, the hypotenuse measures longer than either of its legs. In this case, the hypotenuse is indeed longer than either of the two legs of the triangle (shown in purple).

Also, notice that travel directly along the hypotenuse is indeed the shortest distance between the two locations. The other alternative would be to travel first along the horizontal leg of the triangle 5 degrees and then move vertically 3 degrees, for a total trip of 8 degrees. Thus, the hypotenuse is the shortest path of travel between NASA Dryden and NASA Ames.

Provide students with additional practice using the Pythagorean Theorem by finding the shortest distance (measured in degrees) between other NASA sites.

 Enrichment Activity: To provide students with more practice using the Pythagorean Theorem, use the
latitude and longitude coordinates of various other cities, or, compute distances between state capitals.

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