Question

ABC is a right isosceles triangle with hypotenuse AB M is the midpoint of AB . write a coordinate proof to show that CM is perpendicular to AB

Answer 1

Answer:

Let C(0,0), A(a,0), and B(0,a) be the vertices of the isosceles right triangle. Then the midpoint M of segment AB is (a/2, a/2), by direct use of the midpoint formula. Then segment CM has slope %28a%2F2+-+0%29%2F%28a%2F2+-+0%29+=+1, while segment AB has slope %28a-0%29%2F%280-a%29+=+-1. And because the two segments' slopes have a product of -1, these segments are perpendicular.  

 

You can also use the converse of the Pythagorean theorem on C(0,0), M(a/2, a/2), and A(a,0):