Latitude and Longitude:
A Real Life Example of the Pythagorean Theorem
Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo
Audience: Grades 7-8, algebra I students
Mathematical Topics:
Pythagorean Theorem, coordinate graphing
Materials:
latitude and longitude coordinates of NASA sites
map of U.S.
The Activity:
In the
previous 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 ).
PLEASE NOTE: For simplicity sake, let us make the assumption that the world is flat so that we can apply the distance formula to this problem. Certainly, the earth is round! Thus, the distances computed in this activity will not be as exact as if we were considering the curvature of the earth. However, given that the coordinates of latitude and longitude clearly define a specific location, we will use this opportunity to compute the distance between two locations.
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.
Recall that we are making the assumption that the world is flat so that we can apply the distance formula to this problem. Certainly, the earth is round! Thus, the distances computed in this activity will not be as exact as if we were considering the curvature of the earth. However, given that the coordinates of latitude and longitude clearly define a specific location, we will use this opportunity to compute the distance between two locations.
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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Funded by the
NASA Dryden Flight Research Center
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