Skill: Reflect figures across horizontal and vertical lines

A reflection flips a figure across a line. Each point moves to the opposite side of the line by the same perpendicular distance. The line of reflection is halfway between each point and its image, so the figure keeps the same size, shape, and angle measures.

For a horizontal line $y=k$:

$$(x,y)\longrightarrow (x,2k-y)$$

The $x$-coordinate stays the same, and the $y$-coordinate changes.

For a vertical line $x=h$:

$$(x,y)\longrightarrow (2h-x,y)$$

The $y$-coordinate stays the same, and the $x$-coordinate changes.

Worked example

Reflect triangle $ABC$ across the horizontal line $y=2$.

Let

$$A(1,5),\qquad B(4,3),\qquad C(2,0).$$

Since the line is $y=2$, use

$$(x,y)\longrightarrow (x,2(2)-y)=(x,4-y).$$

Reflect each vertex:

• For $A(1,5)$:

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

• For $B(4,3)$:

$$B'(4,4-3)=(4,1)$$

• For $C(2,0)$:

$$C'(2,4-0)=(2,4)$$

Therefore, the reflected triangle has vertices

$$\boxed{A'(1,-1),\ B'(4,1),\ C'(2,4)}.$$

Check the distances from the line $y=2$: $A$ is $3$ units above it, and $A'$ is $3$ units below it. The same idea works for every point. For a reflection across a vertical line, keep $y$ the same and change the $x$-coordinate using $(x,y)\to(2h-x,y)$.