MHB Symmetry of Points Across y = x Line?

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The discussion focuses on proving that the points (a, b) and (b, a) are symmetric about the line y = x. It establishes that the slope of the line connecting these points is -1, indicating it is perpendicular to y = x, which has a slope of 1. To demonstrate symmetry, it is necessary to show that the midpoint of the segment connecting these points lies on the line y = x. The midpoint is calculated as the arithmetic means of the coordinates of the points A(a, b) and B(b, a). This proof confirms the symmetry of the points across the line y = x.
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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)?
 
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Note that $$m=\frac{a-b}{b-a}=-1$$. 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 $B$.
 
Evgeny.Makarov said:
Note that $$m=\frac{a-b}{b-a}=-1$$. 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?
 
RTCNTC said:
Must I find the midpoint using the given points?
Yes, as I said,
Evgeny.Makarov 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 $B$.
 
Evgeny.Makarov said:
Yes, as I said,

Great. Very informative.
 
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