Question

using coordinates, write a detailed step by step proof that the set of points equidistant from two fixed points, A and B, is the perpendicular bisector of segment AB

Answer 1

Consider the coordinate system shown. Suppose 2 points on the x axis as :

Point A has coordinates ( x₁ , 0 ) and point B has coordinates ( x₂ , 0 ) .

C(x,y) (x2,0) (x1,0)

Now , consider any point C . Let coordinates of C be ( x , y ) . That means x can lie anywhere on the 2D plane.

For point C to lie on perpendicular bisector of AB , length AB should be equal to length BC.

i.e $AB=BC$    ….. Equation (1)

We know distance between two points
(a, b
and
(c, d)
can be calculated using the following formula :

Length = V (a-c)2 + (b-d)2

For equation (1) we get :

$\sqrt{(x_{1}-x)^2+(y)^2}=\sqrt{(x_{2}-x)^2+(y)^2}$

Square both sides :

$(x_{1}-x)^2+(y)^2=(x_{2}-x)^2+(y)^2$

Cutoff common terms -

$(x_{1}-x)^2=(x_{2}-x)^2$

Rearrange as :

$(x_{1}-x)^2-(x_{2}-x)^2=0$

Use the formula :

(a + b) (a-b) = a2-b2

We get :

$(x_1-x+x_2-x)(x_1-x-x_2+x)=0$

$(x_1+x_2-2x)(x_1-x_2)=0$

Neglect the second bracket as it states x₁ = x_{2 .}

_{$x_1+x_2-2x=0$}

_{$x_1+x_2=2x$}

_{$x=\frac{x_1+x_2}{2}$}

From above equation we see that x coordinate of point C is lying in middle point of AB.

As value of y is varied we will get the perpendicular bisector of AB .

_{, y (x1,0) (x2,0)}