# Honors Geometry Companion Book, Volume 1

1.2.2 Midpoint and Distance in the Coordinate Plane (continued)

The Distance Formula could be used to find the distance from P to R . Alternatively, a 2 + b 2 = c 2 could also be used. At first glance it may appear that the Pythagorean Theorem does not apply, but a right triangle can be drawn where the hypotenuse is the line segment from P to R . The legs of the right triangle are horizontal and vertical line segments. Since the legs are horizontal and vertical, the lengths of the legs can be found by subtracting their coordinates (or by counting along the axes). To draw the right triangle, draw a segment to the right from P and up from R . The point at which these two segments meet is the third vertex of the right triangle, marked here as Q . The horizontal leg, named a , extends along the x -axis from − 4 to 2. So, a is 6. The vertical leg, named b , extends along the y -axis from − 2 to 6. So, b is 8. Substitute these values into a 2 + b 2 = c 2 and solve for c , which is the distance from P to R . In this example, a 2 + b 2 = c 2 is used to find the hypotenuse of a right triangle. The baseball field is a square, so there are four right angles, one at each base. It is given that each side of the square is 50 feet. The distance from first base to third base is the hypotenuse of a right triangle where one leg is the length from first base to second base and the second leg is the length from second base to third base. Therefore, the length of each leg is 50. Substitute 50 into a 2 + b 2 = c 2 for a and for b and let the length of the hypotenuse be represented by D .Then, simplify to find D , the distance from first base to third base.

Example 5 Sports Application

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