🎯⭐ INTERACTIVE LESSON

Reflection

Learn step-by-step with interactive practice!

Reflection - Complete Interactive Lesson

Part 1: 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

Section
The Geometric Definition
Coordinate Rules for the Common Mirrors
Reflecting a Whole Figure
Orientation: Why Reflections "Flip"
Composing Two Reflections

🔑 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:

The line $\ell$ is the perpendicular bisector of $\overset{}{P}$ .

📐 Coordinate Rules for Common Mirrors

For the four most common reflection lines in coordinate geometry:

Mirror line	Coordinate rule	What happens
$x$ -axis	$(x, y) \to (x,\, -y)$	Keep , negate

✏️ Reflecting a Whole Figure

Worked Example

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

🔄 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:

Composition	Orientation effect
One reflection	Reverses orientation
Two reflections	Preserves orientation (net effect is a translation or rotation)
Three reflections	Reverses orientation (net effect is a glide reflection)

Check: Identifying Reflections

⚠️ Common Mistakes

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 .

Exit Quiz: Reflection Mastery

\overline{P P^{'}}

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.

\text{dist}(P,\, \ell) \;=\; \text{dist}(P',\, \ell), \qquad \overline{PP'} \perp \ell.

💡 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 $0$ ).

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 $(x, y) \to (x,\, 2k - y)$ .

Vertex	Rule $(x, y) \to (-x,\, y)$	Image
$A(-1, 4)$	$(1,\, 4)$	$A'(1, 4)$
$B(2, 1)$	$(-2,\, 1)$	$B'(-2, 1)$
$C(4, 5)$	$(-4,\, 5)$	$C'(-4, 5)$

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.

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

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.

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).

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.