Points on an angle bisector and points on a perpendicular bisector each have important properties.
Points on an Angle Bisector
The distances from a point on an angle bisector to the two sides forming the angle are equal.
Points on a Perpendicular Bisector
The distances from a point on the perpendicular bisector of a line segment to the two endpoints of the segment are equal.
Example ①
When point P lies on the perpendicular bisector of line segment AB, prove that PA=PB.
Answer: Let M be the midpoint of AB, and let PM⊥AB. In △PAM and △PBM / AM=BM (M is the midpoint) ① / ∠PMA=∠PMB=90° ② / PM=PM (common side) ③ / By SAS, △PAM≅△PBM. Therefore, PA=PB.
When point P lies on the perpendicular bisector of line segment AB, prove that PA=PB.
Answer: Let M be the midpoint of AB, and let PM⊥AB. In △PAM and △PBM / AM=BM (M is the midpoint) ① / ∠PMA=∠PMB=90° ② / PM=PM (common side) ③ / By SAS, △PAM≅△PBM. Therefore, PA=PB.
Key Points
- On a perpendicular bisector ↔ equidistant from the two endpoints
- On an angle bisector ↔ equidistant from the two sides
- These properties are often used in proofs
Practice Problem
- In △ABC, when AB=AC, show that the perpendicular bisector of BC passes through vertex A.
Answers and Explanations
- Answer: Since AB=AC, A is equidistant from B and C. By the converse of the perpendicular bisector property, A lies on the perpendicular bisector of BC.