Reflection over Line y = -x

Understanding Reflection over the Line y = -x

Reflection over the line $y = -x$ is a geometric transformation where each point of an object is mapped to a new position, with the line $y = -x$ acting as a mirror.

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

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, $P'(x', y')$, are determined by the rule:

Thus, the image of point $P(x, y)$ is $P'(-y, -x)$. Notice that the x and y coordinates swap places AND change signs (become their negatives).

Reflecting a Point

Suppose we have point $D(2, 1)$. If point D is reflected over the line $y = -x$, its image, $D'$, is:

Thus, the image of point D is $D'(-1, -2)$.

Let's visualize the reflection of several points over the line $y = -x$:

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:

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 $y = -4x - 2$ 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 $-y$ and every $y$ with $-x$.

Original equation:

Substitute $x \rightarrow -y$ and $y \rightarrow -x$:

Simplify the equation for the image line:

So, the equation of the image of the line $y = -4x - 2$ after reflection over $y=-x$ is $y = -\frac{1}{4}x + \frac{1}{2}$.

Exercises

1. Determine the coordinates of the image of point $S(5, -9)$ 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 $4x + 3y - 2 = 0$ is reflected over the line $y = -x$, determine the equation of its image.

Key Answers

1. The image of point $S(5, -9)$ is $S'(9, -5)$.

   Explanation:

   $$x' = -(-9) = 9$$

   $$y' = -(5) = -5$$

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 $4x + 3y - 2 = 0$ is $4(-y) + 3(-x) - 2 = 0$.

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

   $$4(-y) + 3(-x) - 2 = 0$$

   $$-4y - 3x - 2 = 0$$

   $$3x + 4y + 2 = 0$$