Question

Given that AD is a perpendicular bisector of BC, what steps will prove that A is equidistant from B and C?

Answer 1

To prove that A is equidistant from B and C with AD as the perpendicular bisector of BC, use congruence of triangles ABD and ACD, using the RHS Rule of Congruence to conclude that AB equals AC.

Step-by-step explanation:

The question asks for the steps to prove that point A is equidistant from points B and C given that line segment AD is a perpendicular bisector of BC. In geometry, especially in the study of triangles and congruence, this involves a number of standard steps:

1. Identify triangle ABD and triangle ACD.
2. Since AD bisects BC, BD and CD are equal in length, by definition of bisector.
3. AD is a common side to both triangles ABD and ACD.
4. Angle ADB and angle ADC are right angles because AD is perpendicular to BC.
5. By the Reflexive Property, segment AD is equal to itself.
6. Now, by the Pythagorean theorem or by the RHS (Right angle-Hypotenuse-Side) Rule of Congruence (also known as HL - Hypotenuse-Leg), the triangles ABD and ACD are congruent.
7. Since the triangles are congruent, the corresponding sides AB and AC are equal, which means point A is equidistant from points B and C.