Use indirect proof with geometry What is one way you could determine if a point lies on the perpendicular bisector of a segment?
One way to determine if a point lies on the perpendicular bisector of a segment is by using indirect proof. Here's how you can do it:
1. Assume that the point does not lie on the perpendicular bisector of the segment.
2. Then, draw a line segment from the given point to any point on the perpendicular bisector of the segment.
3. Construct the perpendicular bisector of this new line segment.
4. Consider the two triangles formed by the original segment and the new line segment, and by the new line segment and its bisector.
5. Use the SAS (side-angle-side) congruence postulate or theorem to prove that the two triangles are congruent.
- Show that the two corresponding sides are congruent, i.e., the original segment and the new line segment are congruent due to their common endpoint.
- Show that the included angles are congruent, i.e., the angle formed by the original segment and the new line segment is congruent to the angle formed by the new line segment and its bisector, as they are both right angles.
- Show that the other corresponding sides are congruent, i.e., the length of the segment from the point to the perpendicular bisector is congruent to the length of the segment from the point to the new line segment.
6. Since the triangles are congruent, the corresponding parts of congruent triangles are congruent (CPCTC), so the segment from the point to the perpendicular bisector must be congruent to the segment from the point to the new line segment.
7. However, this would imply that the point lies on the perpendicular bisector, contradicting our original assumption.
8. Therefore, our assumption was incorrect, and the point must indeed lie on the perpendicular bisector of the segment.