Math Core

Look at your reflection in a still lake and you see a flipped copy of the world above it. A reflection in geometry works the same way: it flips a figure over a line, like folding the paper along that line.

Reflecting over a line

Definition

Reflection

A reflection flips a figure over a line called the line of reflection. Each point and its image are the same distance from the line, on opposite sides, and the segment joining them is perpendicular to the line.

A point that sits on the line of reflection doesn't move at all. Its image is itself.

Reflecting across the axes

The x-axis and y-axis are the most common lines of reflection, and each one has a simple rule.

Reflection rules

line of reflection	rule	example
x-axis	$(x, y) \rightarrow (x, -y)$	$(3, 5) \rightarrow (3, -5)$
y-axis	$(x, y) \rightarrow (-x, y)$	$(3, 5) \rightarrow (-3, 5)$

Worked example: Across the x-axis

Reflect triangle $ABC$ with $A(1, 2)$ , $B(5, 2)$ and $C(2, 5)$ across the x-axis.

Keep each x-coordinate and take the opposite of each y-coordinate.

\begin{aligned} A(1, 2) &\rightarrow A'(1, -2) \\ B(5, 2) &\rightarrow B'(5, -2) \\ C(2, 5) &\rightarrow C'(2, -5) \end{aligned}

Triangle ABC reflected across the x-axis to triangle A′B′C′.Open in grapher →

Point $A$ was 2 units above the x-axis, and $A'$ is 2 units below it. Every vertex is the same distance from the axis as before.

Common mistake

Reflecting across the x-axis changes the y-coordinate, not the x-coordinate. That feels backward at first. Think about it this way: flipping over a horizontal line moves points up or down, and up/down is the y-direction.

Worked example: Across the y-axis

Reflect triangle $DEF$ with $D(-5, 1)$ , $E(-2, 1)$ and $F(-2, 4)$ across the y-axis.

Take the opposite of each x-coordinate and keep each y-coordinate: $D'(5, 1)$ , $E'(2, 1)$ and $F'(2, 4)$ .

Triangle DEF reflected across the y-axis to triangle D′E′F′.Open in grapher →

Reflecting across other lines

For a vertical line like $x = 1$ or a horizontal line like $y = -2$ , count. Find how far the point is from the line, then go the same distance on the other side.

Worked example: Across the line x = 1

Reflect triangle $ABC$ with $A(-2, 3)$ , $B(0, -1)$ and $C(-3, -1)$ across the vertical line $x = 1$ .

A vertical line only changes x-coordinates. The y-coordinates stay the same.

$A(-2, 3)$ is 3 units left of $x = 1$ . Go 3 units right of the line: $A'(4, 3)$ .
$B(0, -1)$ is 1 unit left of the line. Go 1 unit right: $B'(2, -1)$ .
$C(-3, -1)$ is 4 units left of the line. Go 4 units right: $C'(5, -1)$ .

Triangle ABC reflected across the line x = 1.Open in grapher →

What stays the same

A reflection is a rigid motion, so lengths and angle measures don't change. The image is an exact copy of the pre-image.

One thing does change: orientation. In the first example, $A \to B \to C$ goes counterclockwise, but $A' \to B' \to C'$ goes clockwise. The image is a mirror image, the way your right hand looks like a left hand in a mirror.

Tip

Check a reflection by folding. If you folded the graph along the line of reflection, each vertex should land exactly on its image.

Practice

Practice 1

Reflect the point $(7, -3)$ across the x-axis. What is the image?

Enter a point like (2, -3)

Practice 2

Reflect the point $(-6, -2)$ across the y-axis. What is the image?

Enter a point like (2, -3)

Practice 3

A point in Quadrant II is reflected across the x-axis. In which quadrant is the image?

Practice 4

Reflect the point $(5, 4)$ across the horizontal line $y = 1$ . What is the image?

Enter a point like (2, -3)

Practice 5

Reflect the point $(-3, 6)$ across the vertical line $x = 2$ . What is the image?

Enter a point like (2, -3)

Practice 6

The vertices of triangle $ABC$ go clockwise in the order $A \to B \to C$ . The triangle is reflected across the y-axis. In which direction do $A' \to B' \to C'$ go?

Practice 7

Reflect the point $(2, -5)$ across the x-axis, and then reflect that image across the y-axis. Where does the point end up?

Enter a point like (2, -3)

Practice 8

The points $A(-4, 2)$ and $A'(6, 2)$ are reflections of each other across a vertical line $x = k$ . What is $k$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.