Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base .

Given: An Isosceles Triangle ABC with a vertex at B.

Midpoint M of the base AC.

To Prove: BM is perpendicular to AC.  

Proof:

Let the coordinates of the points of the isosceles triangle be given as:

A =  (-k, 0)

Vertex, B =  (0,a)  

C = (k, 0)

Midpoint, M = (0,0)

Slope of the base segment, AC:

Slope of the base segment,  AC=

Slope of the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base, BM.

 = Undefined

Two lines are perpendicular if the gradient of one is a negative reciprocal of the other.

Since is a negative reciprocal of 0 for arbitrary values of a, BM and AC are perpendicular.

This concludes the proof.

answer: a and b

step-by-step explanation: