Need help writing a coordinate proof

[cenglish]

Question:
Triangle ABC is a right isosceles triangle with hypotenuse AB. M is the midpoint of Line AB. Write a coordinate proof to prove that Line CM is perpendicular to Line AB.

 

[Mentallic]

This problem is very simple. Draw out the diagram, label all the points, and appropriately label the sides that are equal to each other. Tell us what you get.

 

[cenglish]

This problem is very simple. Draw out the diagram, label all the points, and appropriately label the sides that are equal to each other. Tell us what you get.

I got: The midpoint of Line AB is (a,0). The slope of Line CM is undefined and the slope of Line AB is 0. Therefore, Line AB is perpendicular to Line CM.

 

[Mentallic]

I misinterpreted "coordinate proof" as being something else.

I got: The midpoint of Line AB is (a,0). The slope of Line CM is undefined and the slope of Line AB is 0. Therefore, Line AB is perpendicular to Line CM.

What are the coordinates of A,B,C? If you're going to claim these things, you have to show the calculations you've done to prove it. Why does CM have an undefined slope for example?

Or ideally, you should start by drawing the triangle in the simplest way possible. Since it's a right-triangle, use the fact that the x and y axes intersect at the origin at right angles to each other. That is, let
A = (0,a)
B = (a,0)
C = (0,0)

Notice that B is at position (a,0) because the lengths of AC = BC.

 

[HallsofIvy]

Set up a coordinate system with the origin at the right angle, the positive x and y axes along the two legs. Then one vertex is at (a, 0) and another at (0, a) for some number a. What are the coordinates of M? Show that the slope of the line from (0, 0) to M is the negative reciprocal of the line from (0, a) to (a, 0).