Reflection over Line y = x

Understanding Reflection over the Line y = x

Reflection over the line $y = x$ is a geometric transformation that maps every point of an object to a new position with the line $y = x$ acting as a mirror.

The distance from the original point to the mirror line is equal to the distance from the image point to the mirror line, and the line connecting the original point to its image will be perpendicular to the line $y = x$.

Rule for Reflection over the Line y = x

If a point $P(x, y)$ is reflected over the line $y = x$, its image's coordinates, $P'(x', y')$, will follow a simple rule:

Thus, the image of point $P(x, y)$ is $P'(y, x)$. Notice that the x and y coordinates swap positions.

Reflecting a Point

Suppose we have point $A(1, 4)$. If point A is reflected over the line $y = x$, its image, $A'$, can be determined by swapping its coordinates:

The original x-coordinate is 1, becoming the new y-coordinate.

The original y-coordinate is 4, becoming the new x-coordinate.

Thus, the image of point A is $A'(4, 1)$.

Let's visualize this along with some other points:

Reflecting a Triangle

Determine the image of triangle ABC with vertices $A(-2,4)$, $B(3,1)$, and $C(-3,-1)$ reflected over the line $y=x$.

To reflect the triangle, we reflect each of its vertices over the line $y=x$:

1. Point $A(-2, 4)$: Its image is $A'(4, -2)$.
2. Point $B(3, 1)$: Its image is $B'(1, 3)$.
3. Point $C(-3, -1)$: Its image is $C'(-1, -3)$.

The image triangle $A'B'C'$ is formed by connecting the points $A'(4, -2)$, $B'(1, 3)$, and $C'(-1, -3)$.

Reflecting a Line Equation

If a line has the equation $y = 2x + 3$ is reflected over the line $y=x$, determine the equation of its image.

To find the equation of the image, we use the rule $x' = y$ and $y' = x$. This means we replace every $x$ in the original equation with $y$ (or $y'$) and every $y$ with $x$ (or $x'$).

Original equation:

Substitute $x \rightarrow y$ and $y \rightarrow x$ (using $x$ and $y$ for the new variables for simplicity):

This is the equation of the image line. Usually, we rewrite this equation in the form $y$ as a function of $x$:

So, the equation of the image of the line $y = 2x + 3$ after reflection over $y=x$ is $y = \frac{1}{2}x - \frac{3}{2}$.

Exercises

1. Determine the coordinates of the image of point $M(-4, 7)$ if it is reflected over the line $y = x$!
2. Determine the image of triangle ABC with vertices $A(1, -2)$, $B(2, 3)$, and $C(-2, 0)$ reflected over the line $y = x$.
3. If a line has the equation $y = -3x + 5$ is reflected over the line $y = x$, determine the equation of its image.

Key Answers

1. The image of point $M(-4, 7)$ is $M'(7, -4)$.

   Explanation: The x and y coordinates are swapped: $x' = 7$, $y' = -4$.

2. The coordinates of the image triangle $A'B'C'$ are:

   • $A'(-2, 1)$ (from $A(1, -2)$)
   • $B'(3, 2)$ (from $B(2, 3)$)
   • $C'(0, -2)$ (from $C(-2, 0)$)

3. The equation of the image of the line $y = -3x + 5$ is $x = -3y + 5$.

   Explanation: Substitute $x \rightarrow y$ and $y \rightarrow x$ into the original equation:

   $$x = -3y + 5$$

   If converted to the form $y = mx+c$:

   $$3y = -x + 5$$

   $$y = -\frac{1}{3}x + \frac{5}{3}$$