Let's place a right isosceles triangle in the coordinate plane. To do that, we place the legs on the coordinate axes.
By applying the Midpoint Formula we find M, the midpoint of the hypotenuse.
M(x_1+x_2/2,y_1+y_2/2) &= M(a+0/2,0+a/2)
&= M(a/2,a/2)
Next, we draw the segment CM.
Our mission is to prove that CM⊥ AB. To prove this, we will find the equation of the lines containing each segment, then we will take a look at their slopes.
Finding the Lines Containing the Segments
The line containing CM passes through the origin and through M. Therefore, we find its slope as follows.
m_1 = a2-0/a2-0 = 1
Since the y-intercept is 0, the equation of this line is y_1=x. The second line passes through A and B, so we find its slope as follows.
m_2 = a-0/0-a = -1
This line has y-intercept a and then its equation is y_2=- x+a.
Comparing the Slopes
From the previous part, we obtained that the slope of the line containing CM is 1, while the slope of the line containing AB is -1. This leads us to the following relation.
m_1* m_2 = 1 (-1) = -1
Therefore, the slopes of the lines are reciprocals. Consequently, the lines are perpendicular. This implies that CM⊥ AB, which is what we wanted to prove.
