Reflections

By the end of this lesson, you will be able to reflect figures over horizontal lines, vertical lines, and the lines $y = x$ and $y = -x$, apply the coordinate rules for each reflection, and distinguish reflections from other rigid motions.

The Big Idea

A reflection flips a figure over a line of reflection, creating a mirror image. The line of reflection is the perpendicular bisector of every segment connecting a pre-image point to its image. Like translations, reflections are isometries — they preserve distance and angle measure. Unlike translations, they reverse orientation.

Definition

A reflection over line $\ell$ maps every point $P$ to point $P'$ such that:

• $\ell$ is the perpendicular bisector of $\overline{PP'}$
• Points on $\ell$ map to themselves (they are fixed points)

Coordinate Rules for Common Reflections

Line of reflection	Rule	Effect
$x$-axis	$(x, y) \to (x, -y)$	Negate $y$
$y$-axis	$(x, y) \to (-x, y)$	Negate $x$
$y = x$	$(x, y) \to (y, x)$	Swap $x$ and $y$
$y = -x$	$(x, y) \to (-y, -x)$	Swap and negate both
$x = a$ (vertical line)	$(x, y) \to (2a - x, y)$	Reflect horizontally
$y = b$ (horizontal line)	$(x, y) \to (x, 2b - y)$	Reflect vertically

Example 1 — Over the $x$-axis

Reflect $\triangle ABC$: $A(2, 3)$, $B(5, 1)$, $C(4, -2)$ over the $x$-axis.

$$A' = (2, -3), \quad B' = (5, -1), \quad C' = (4, 2)$$

Example 2 — Over $y = x$

Reflect $P(3, -7)$ over $y = x$.

$$P' = (-7, 3)$$

Example 3 — Over a Horizontal Line

Reflect $A(5, 4)$ over the line $y = 1$.

Using $(x, 2b - y)$ with $b = 1$:

$$A' = (5, 2(1) - 4) = (5, -2)$$

Verification: midpoint of $\overline{AA'}$ is $(5, \frac{4 + (-2)}{2}) = (5, 1)$, which lies on $y = 1$ ✓

Example 4 — Over a Vertical Line

Reflect $B(7, 3)$ over $x = 2$.

Using $(2a - x, y)$ with $a = 2$:

$$B' = (2(2) - 7, 3) = (-3, 3)$$

Properties of Reflections

1. Isometry: preserves distance and angle