Reflection over the Negative Diagonal Line

Understanding Reflection over the Negative Diagonal Line

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 Negative Diagonal Line

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:

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

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

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