If A and B are two points in the plane the perpendicular bisector of line segment AB is the set of all points equidistant from A and B?

Answer 1

Let us consider a line l such that it is the perpendicular bisector of line segment AB and the line intersects at point C.

Let us take any point on line l(say, D). Join A to D and B to D.

Now we have two triangles ACD and BCD.

Now, in triangles ACD & BCD, we have

CD = CD (Common side)

�ACD = �BCD (Right angle)

AC = BC (Since l bisects AB)

According to Side-Angle-Side criteria: Both triangles are congruent.

Since both triangles are congruent, therefore AD = BD.

So, l is the set of all points equidistant from A & B.

Hence proved.