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The Distance Between Two Points Mr. Dick Gill Dr. Marcia Tharp Introduction When the Wizard of Oz finally gave the scarecrow his brain the first words that the scarecrow spoke were: “The sum of the squares of the legs of an isosceles right triangle is equal to the square of the hypotenuse.” He was quoting a special case of the Pythagorean Theorem. The famous Greek mathematician Pythagoras is credited with the discovery illustrated on the next slide concerning the three sides of any right triangle (a triangle with one angle equal to 90 degrees). Pythagoras was a philosopher and a mystic as well as a mathematician. For more on Pythagoras go to: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html The hypotenuse of a right triangle is always the longest side, the side opposite the right angle. The legs are the other two sides. In the Pythagorean Theorem, it is customary to use c for the hypotenuse. For an attractive and simple demo on why this theorem works go to: http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/PythagorasTheorem.gif Consider the following triangle with legs of 6 inches and 8 inches. Find the hypotenuse. See next page. Solution a 2 b 6 2 8 2 c 2 c 2 36 64 c 2 2 100 c 100 c 10 c 2 The hypotenuse then would be 10 inches long. Further work with the Pythagorean Theorem will require a review of square roots. As indicated above, the square root of 100 is 10 because 102 = 100. 2 81 9 since 9 81 2 64 8 since 8 64 2 49 7 since 7 49. Looking at the roots above, you may conclude that would be a number between 8 and 7. In fact, is approximately 7.75. For the rest of this unit you will need a calculator that does square roots. For more about square roots go to: http://forum.swarthmore.edu/~sarah/hamilton/ham.squa reroots.html Example 5 Find the missing side in each of the following triangles. To view the exercise please click here. • Once you have completed the exercise click to the next slide to check your answers. Solutions: a. 6 2 1 22 x 2 36 144 x2 180 x2 x b. 1 8 0 1 3.4 2 x 2 72 92 x2 49 81 x2 32 x 3 2 5 .6 6 c. 1 12 x 2 1 52 1 2 1 x 2 2 2 5 x2 105 x 1 0 5 1 0.2 5 Now lets see how this works in a coordinate system. 1. Click on the link below to see how one finds the distance between two points in a coordinate system. 2. Experiment by changing the coordinates of the point P. 3. Write a paragraph to explain how and why this works to a fellow student. • Link • Link to index