5.1.1 Perpendicular and Angle Bisector Theorems (continued) Example 3 Writing Equations of Bisectors in the Coordinate Plane In this example, the equation of a perpendicular bisector is written. Remember, the equation of a line can be found by using either two points that the line passes through or one point that the line passes through and the slope of the line.
The coordinates of the endpoints of PQ are given. Neither of these points can be used to write the equation of the perpendicular bisector of PQ because the perpendicular bisector does not pass through the segment’s endpoints. Rather, since the perpendicular bisector of PQ bisects PQ , it must pass through the midpoint of PQ . So, find coordinates of the midpoint of PQ by using the Midpoint Formula: M (2, 1). The perpendicular bisector of PQ must be perpendicular to PQ . Use this fact to find the slope of the perpendicular bisector. First, find the slope of PQ . Then, find the slope of the line perpendicular to PQ . The slope of PQ is − 1. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is 1. Now that a point on the perpendicular bisector is known, M (2, 1), and the slope is known, m = 1, use point-slope form to write the equation of the perpendicular bisector.
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