Show that the line containing the point (a, b) and (b, a) is perpendicular to the line y = x.
Also show that the midpoint of (a, b) and (b, a) lies on the line y = x.
Hello, symmetry!
I must assume that you are familiar with the basic concepts needed:
. . slopes, perpendicular slopes, midpoints.
If you are not, learn or re-learn the formulas and concepts.
Show that the line containing the points $\displaystyle (a, b)$ and $\displaystyle (b, a)$
is perpendicular to the line: $\displaystyle y = x$
The slope of the line through $\displaystyle (a,b)$ and $\displaystyle (b,a)$ is: .$\displaystyle m_1 \:=\:\frac{a-b}{b - a} \:=\:-1$
The slope of the line $\displaystyle y\:=\:x$ is: .$\displaystyle m_2 = 1$
Since $\displaystyle m_1\!\cdot\!m_2 \:=\:(-1)(1) \:=\:-1,\;\;m_1 \perp m_2$
Show that the midpoint of $\displaystyle (a, b)$ and $\displaystyle (b, a)$ lies on the line $\displaystyle y \:= \:x$
The midpoint is: .$\displaystyle \left(\frac{a+b}{2},\,\frac{a+b}{2}\right)$
The two coordinates of the midpoint are equal.
. . Therefore, they satisfy the equation $\displaystyle y \:=\:x$