Reflection over a Horizontal Line

Equal Distances on Either Side of a Horizontal Mirror

Reflection over the horizontal line $y = h$ moves each point of an object to its reflected position. The line $y = h$ is the mirror line.

The vertical distance from the original point to the mirror line is equal to the vertical distance from the image point to the mirror line. The $x$-coordinate of the point does not change.

Rule for Reflection over a Horizontal Line

If a point $P(x, y)$ is reflected over the line $y = h$, its image's coordinates, $P'(x', y')$, are determined by the rule:

The image of point $P(x, y)$ is $P'(x, 2h - y)$. The $x$-coordinate remains the same, while the $y$-coordinate changes according to its distance from the line $y=h$.

Reflecting a Point over a Horizontal Line

Determine the image of point $P(3,2)$ by reflection over the line $y=3$.

In this case, $x=3$, $y=2$, and $h=3$.

Using the rule $P'(x, 2h - y)$:

The image of point $P(3,2)$ is $P'(3,4)$.

The diagram shows the equal vertical distances.

Exercises

Each problem reflects over the horizontal line $y=h$. The last one starts from the image, so the same formula runs in reverse.

1. Determine the image of point $P(5,-4)$ by reflection over the line $y=-2$.
2. A point $Q(-1, -3)$ is reflected over the line $y=0$ ($x$-axis). Determine the coordinates of its image!
3. The image of a point $R(x,y)$ after reflection over the line $y=4$ is $R'(-2,5)$. Determine the coordinates of point $R$!

Solutions

1. Given $P(5,-4)$ and the mirror line $y=-2$. So $x=5, y=-4, h=-2$.

   $$x' = x = 5$$

   $$\begin{aligned} y' &= 2h - y \\ &= 2(-2) - (-4) \\ &= -4 + 4 \\ &= 0 \end{aligned}$$

   Thus, the image of point $P$ is $P'(5,0)$.

2. Given $Q(-1, -3)$ and the mirror line $y=0$. So $x=-1, y=-3, h=0$.

   $$x' = x = -1$$

   $$\begin{aligned} y' &= 2h - y \\ &= 2(0) - (-3) \\ &= 0 + 3 \\ &= 3 \end{aligned}$$

   Thus, the image of point $Q$ is $Q'(-1,3)$.

3. Given the image $R'(-2,5)$ and the mirror line $y=4$. So $x'=-2, y'=5, h=4$.

   We know $x' = x$ and $y' = 2h - y$.

   From $x' = x$, then $x = -2$.

   From $y' = 2h - y$, then $5 = 2(4) - y$.

   $$5 = 8 - y$$

   $$y = 8 - 5$$

   $$y = 3$$

   The coordinates of point $R$ are $R(-2,3)$.