5.1.1 Perpendicular and Angle Bisector Theorems (continued)
The measure of a line segment is determined here using the Perpendicular Bisector Theorem. In the figure, PL is given to be perpendicular to MN . Additionally, PL bisects MN since MP = PN . Therefore, PL is a perpendicular bisector of MN . So, the Perpendicular Bisector Theorem can be applied. The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment must be equidistant from the endpoints of that segment. Here, L is a point on perpendicular bisector PL . So, L must be equidistant from the endpoints of MN , M and N , by the Perpendicular Bisector Theorem. It follows that ML = NL . It is given that NL = 7. Thus, ML = 7 as well. Here, PL is not given to be perpendicular to MN . However, it is given that ML ≅ NL . It follows that x = 8 since NL = 8 and ML = x . Since x = 8 and MP = x /2, it follows that MP = 8/2 = 4. It would be easy to assume that since MP = 4, PN = 4 as well, but first the fact that MP = PN must be confirmed since it is not given. Use the Converse of the Perpendicular Bisector Theorem to confirm that MP = PN .
Example 2 Using Angle Bisector Theorems
By the Angle Bisector Theorem, if a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Notice that the distance between the point and the sides of the angle is the length of a line segment perpendicular to the side of the angle. This is the shortest distance.
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