Reflection

Reflection: Mirror Images Across a Line

Focus: Flip a figure across a straight line so that each point and its image are equidistant from the line on opposite sides.

Topics in This Lesson

🔑 Big Idea: A reflection is the rigid transformation that reverses orientation. It is the only one of the three basic isometries that turns a left-handed figure into a right-handed one.

What You'll Master

• Apply coordinate rules for reflections over the $x$-axis, $y$-axis, line $y = x$, and line $y = -x$
• Use the perpendicular-bisector property to reflect a point across any line
• Predict how vertex orientation changes after a reflection
• Recognize that two reflections compose to a translation or rotation

Entrance Quiz: Reflection Readiness

🪞 The Geometric Definition

A reflection across a line $\ell$ takes a point $P$ to the point $P'$ such that:

1. The line $\ell$ is the perpendicular bisector of $\overline{PP'}$.
2. If $P$ already lies on $\ell$, then $P' = P$ (points on the mirror are fixed).

That is the only property you need — every coordinate rule below is a consequence of it.

💡 Fixed points of a reflection form a whole line (the mirror itself). Compare to translations (no fixed points unless $\vec{v} = \vec{0}$) and rotations (exactly one fixed point unless the angle is $0$).

📐 Coordinate Rules for Common Mirrors

For the four most common reflection lines in coordinate geometry:

Where do the swap rules come from?

For reflection over $y = x$: the line $y = x$ has slope $1$. The perpendicular from $(a, b)$ to this line has slope $-1$, and the foot of the perpendicular is at $\big(\tfrac{a+b}{2},\, \tfrac{a+b}{2}\big)$. Doubling the displacement from $(a, b)$ to the foot gives image $(b, a)$ — coordinates swap.

🔑 Vertical mirrors ($x = h$): rule is $(x, y) \to (2h - x,\, y)$. Horizontal mirrors ($y = k$): rule is $(x, y) \to (x,\, 2k - y)$.

✏️ Reflecting a Whole Figure

Worked Example

Reflect triangle $A(-1,\, 4),\; B(2,\, 1),\; C(4,\, 5)$ over the $y$-axis.

Plot $A',\, B',\, C'$ and connect them. The image triangle has the same side lengths and angles, but if the original was traversed counterclockwise, the image will be traversed clockwise — that is orientation reversal.

🔄 Orientation: Why Reflections "Flip"

Every triangle in the plane has a signed orientation:

• Positive (counterclockwise) if the vertices $A,\, B,\, C$ are listed in counterclockwise order
• Negative (clockwise) if they are listed in clockwise order

A single reflection multiplies the orientation by $-1$. So:

💡 Diagnostic trick: if you see a transformation that preserves all distances but flips the orientation, it must involve an odd number of reflections.

Check: Identifying Reflections

⚠️ Common Mistakes

1. Negating both coordinates for every reflection. Negating both $x$ and $y$ is a $180°$ rotation, not a reflection over an axis. Reflection over the $x$-axis negates only $y$; reflection over the $y$-axis negates only $x$.

2. Forgetting that points on the mirror stay put. If a vertex lies on the line of reflection, the image equals the original — don't accidentally move it.

3. Mixing up $y = x$ and $y = -x$. Reflection over $y = x$ is $(x, y) \to (y, x)$ (just swap). Reflection over $y = -x$ is $(x, y) \to (-y, -x)$ (swap and negate both).

4. Drawing the image in the same vertex order. After a reflection, the orientation reverses. If $A \to B \to C$ went counterclockwise originally, $A' \to B' \to C'$ now goes clockwise.

Exit Quiz: Reflection Mastery