Reflections

What Is a Reflection?

A reflection flips a shape over a line called the mirror line. The image looks like the shape has been mirrored.

The image is the same size and shape — it’s congruent to the original.

Each point and its image are the same distance from the mirror line.

Any points lying on the mirror line stay where they are. These are invariant points.

Mirror Line Facts

Vertical mirror line $\to$ equation: $x = k$

Horizontal mirror line $\to$ equation: $y = k$

Diagonal lines:

Positive gradient: $y = x$

Negative gradient: $y = -x$

How to Reflect a Shape

Step 1: Draw or identify the mirror line.

Step 2: From each vertex of the shape, measure the perpendicular distance to the mirror line.

Step 3: Reflect each point the same distance on the other side of the mirror line.

Step 4: Join the reflected points to form the image. Label the new shape (e.g. A′B′C′A'B'C').

Example

Question: Reflect triangle $A(2, 3), B(4, 3), C(4, 1)$ in the line $y = -1$.

Solution:

$A(2, 3)$ is 1 unit above the line $y = -1$, so it reflects to $A'(2, 1)$.

$B(4, 3)$ is 1 unit above $y = 2$, so $B'(4, 1)$.

$C(4, 1)$ is 1 unit below $y = 2$, so $C'(4, 3)$.

Reflected coordinates: $A'(2, 1), B'(4, 1), C'(4, 3)$

Example

Question: A shape has vertices at: $(0, 2), (2, 2), (1, 4)$. Reflect it in the line $x = 1$.

Step-by-step:

1. Draw the mirror line $x = 1$ (a vertical line through $x = 1$).
2. Reflect each point:
   • $(0, 2)$ is 1 unit left of the line $\to$ reflected to $(2, 2)$
   • $(2, 2)$ is 1 unit right of the line $\to$ reflected to $(0, 2)$
   • $(1, 4)$ is on the mirror line $\to$ stays at $(1, 4)$
3. Connect the new points to form the reflected triangle.

Reflected coordinates: $(2, 2), (0, 2), (1, 4)$

Describing a Reflection

To describe a reflection:

• Say it is a reflection
• Give the equation of the mirror line

Example: “Shape A has been reflected in the line $y = -1$”

Reversing a Reflection

A reflection is its own opposite:

• Reflect a shape back in the same line to return it to its original position.

Example: A shape is reflected in $x = 3$. To reverse it: reflect it again in $x = 3$.