1.2.2 Midpoint and Distance in the Coordinate Plane (continued) Example 3 Using the Distance Formula
If two points, ( x 1 , y 1 ) and ( x 2 , y 2 ), lie on a coordinate plane, the Distance Formula can be used to find the distance between the two points.
In this example, the lengths of two line segments are found and then compared. Since these two segments lie in a coordinate plane, use the Distance Formula to find the distance between the endpoints, which is equivalent to the length of the segment. First identify the coordinates of each endpoint, then substitute these coordinates into the Distance Formula. As with the Midpoint Formula, it doesn’t matter which of the two points is identified as ( x 1 , y 1 ). However, do be consistent. For example, if x 1 is the x -coordinate of A then y 1 must be the y -coordinate of A.
Example 4 Finding Distances in the Coordinate Plane
The Pythagorean Theorem states that in any right triangle, a 2 + b 2 = c 2 , where a and b are the triangle’s legs and c is the hypotenuse (the side opposite of the right angle). The distance between two points in a coordinate plane can be found using a 2 + b 2 = c 2 .
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