# Thread: Points (a, b) and (b, a)

1. ## Points (a, b) and (b, a)

Show that the line containing the points (a, b) and (b, a),
where a does not equal 0, is perpendicular to the line y = x.

Also, show that the midpoint of (a, b) and (b, a) lies on the line y = x.

2. Originally Posted by magentarita
Show that the line containing the points (a, b) and (b, a),
where a does not equal 0, is perpendicular to the line y = x.

Also, show that the midpoint of (a, b) and (b, a) lies on the line y = x.
The slope of $\displaystyle y=x$ is 1.

You'll need to find the slope of the segment between (a, b) and (b, a).

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{a-b}{b-a}=\frac{-(b-a)}{b-a}=-1$

Since the slopes are negative reciprocals of each other, the lines are perpendicular.

$\displaystyle y=x$ is the linear identity function. this means the x and y values are identical.

The midpoint between (a, b) and (b, a) can be found using the midpoint formula:

$\displaystyle M=\left(\frac{a+b}{2}, \frac{b+a}{2}\right)$

Since both x and y values of the midpoint are identical, then the midpoint has to lie on the identity function $\displaystyle y=x$

3. ## Again...

Originally Posted by masters
The slope of $\displaystyle y=x$ is 1.

You'll need to find the slope of the segment between (a, b) and (b, a).

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{a-b}{b-a}=\frac{-(b-a)}{b-a}=-1$

Since the slopes are negative reciprocals of each other, the lines are perpendicular.

$\displaystyle y=x$ is the linear identity function. this means the x and y values are identical.

The midpoint between (a, b) and (b, a) can be found using the midpoint formula:

$\displaystyle M=\left(\frac{a+b}{2}, \frac{b+a}{2}\right)$

Since both x and y values of the midpoint are identical, then the midpoint has to lie on the identity function $\displaystyle y=x$
Another educational reply.