Symmetry of Points Across y = x Line?

[mathdad]

Show that the points (a, b) and (b, a) are symmetric about the line y = x.

Solution:

Let m = slope

m = (a - b)/(b - a)

I know the slope of y = x is 1.

Must I now show that the line y = x passes through the midpoint? If so, how is this done when the slope is not a number (as in this example)?

 

[Evgeny.Makarov]

Note that \(\displaystyle m=\frac{a-b}{b-a}=-1\). Therefore, the line connecting $A(a,b)$ and $B(b,a)$ is perpendicular to the graph of $y=x$ (it is known that the product of slopes of perpendicular lines is $-1$). It is left to show that the intersection point of the two lines divides the segment $AB$ in half. You can prove this by showing that the middle of $AB$ lies on $y=x$. Recall that the coordinates of the middle of $AB$ are the arithmetic means of the corresponding coordinates of $A$ and $B$.

 

[mathdad]

Note that \(\displaystyle m=\frac{a-b}{b-a}=-1\). Therefore, the line connecting $A(a,b)$ and $B(b,a)$ is perpendicular to the graph of $y=x$ (it is known that the product of slopes of perpendicular lines is $-1$). It is left to show that the intersection point of the two lines divides the segment $AB$ in half. You can prove this by showing that the middle of $AB$ lies on $y=x$. Recall that the coordinates of the middle of $AB$ are the arithmetic means of the corresponding coordinates of $A$ and $B$.

Must I find the midpoint using the given points?

 

[Evgeny.Makarov]

Must I find the midpoint using the given points?

Yes, as I said,

You can prove this by showing that the middle of $AB$ lies on $y=x$. Recall that the coordinates of the middle of $AB$ are the arithmetic means of the corresponding coordinates of $A$ and $B$.

 

[mathdad]

Yes, as I said,

Great. Very informative.