Use indirect proof with algebr What is one way you could determine if a point lies on the perpendicular bisector of a segment?
To determine if a point lies on the perpendicular bisector of a segment, you can use an indirect proof. Here's a step-by-step explanation:
1. Let's assume that the point P does not lie on the perpendicular bisector of the segment AB.
2. The perpendicular bisector of a segment divides it into two equal halves, and any point on the perpendicular bisector is equidistant from the endpoints of the segment.
3. Since we assumed that point P does not lie on the perpendicular bisector, there are two possibilities:
a) Either point P lies on one side of the bisector, closer to one endpoint of the segment, or
b) Point P lies on the other side of the bisector, closer to the other endpoint of the segment.
4. Now, consider the distances from point P to the two endpoints of the segment AB. Let's call them PA and PB.
a) If point P is closer to one endpoint, say A, then PA < PB.
b) If point P is closer to the other endpoint, say B, then PB < PA.
5. In either case, the distances PA and PB are not equal, which contradicts the definition of a point lying on the perpendicular bisector.
6. Therefore, our initial assumption that point P does not lie on the perpendicular bisector of the segment AB must be false.
7. Hence, we conclude that point P does indeed lie on the perpendicular bisector of the segment AB.
By following this indirect proof, you can determine if a point lies on the perpendicular bisector of a segment.