Transformations and Symmetry

A transformation moves or resizes a shape according to a specific rule. There are four fundamental types: translations (slides), reflections (flips), rotations (turns), and dilations (resizing). Together they form the backbone of geometric reasoning and show up on virtually every standardized math test — the SAT, GED, ACT, and Common Core assessments all test these concepts.

Beyond the classroom, transformations are the math behind tiling patterns, graphic design, architecture, and computer animation. Any time you see a repeating wallpaper pattern, a symmetrical logo, or a scaled blueprint, transformations are at work.

Translations (Slides)

A translation slides every point of a figure the same distance in the same direction. The shape does not rotate or flip — it simply moves.

The rule for a translation is:

$(x, y) \to (x + a,\ y + b)$

where $a$ is the horizontal shift and $b$ is the vertical shift. Positive $a$ moves right, negative moves left. Positive $b$ moves up, negative moves down.

A translation preserves the size and shape of the figure. All side lengths, angles, and orientation stay the same.

Example 1: Translate a triangle

Translate the triangle with vertices $A(1, 2)$, $B(4, 2)$, $C(3, 5)$ by the rule $(x, y) \to (x + 3,\ y - 1)$.

Apply the rule to each vertex:

$A(1, 2) \to A'(1 + 3,\ 2 - 1) = A'(4, 1)$

$B(4, 2) \to B'(4 + 3,\ 2 - 1) = B'(7, 1)$

$C(3, 5) \to C'(3 + 3,\ 5 - 1) = C'(6, 4)$

Answer: The translated triangle has vertices $A'(4, 1)$, $B'(7, 1)$, $C'(6, 4)$. Every point moved 3 units right and 1 unit down.

Reflections (Flips)

A reflection flips a figure across a line (called the line of reflection), creating a mirror image. The reflected figure is the same size and shape, but its orientation is reversed — like reading text in a mirror.

The three most common reflections on the coordinate plane are:

Reflect over the x-axis:

$(x, y) \to (x, -y)$

The x-coordinate stays the same; the y-coordinate changes sign.

Reflect over the y-axis:

$(x, y) \to (-x, y)$

The y-coordinate stays the same; the x-coordinate changes sign.

Reflect over the line $y = x$:

$(x, y) \to (y, x)$

The coordinates swap places.

Example 2: Reflect a point over both axes

Reflect the point $(3, 4)$ over the x-axis and over the y-axis.

Over the x-axis: $(x, y) \to (x, -y)$

$(3, 4) \to (3, -4)$

Over the y-axis: $(x, y) \to (-x, y)$

$(3, 4) \to (-3, 4)$

Answer: The reflection over the x-axis gives $(3, -4)$. The reflection over the y-axis gives $(-3, 4)$.

Reflections on the Coordinate Plane

The diagram below shows a triangle and its reflection over the y-axis. Notice how each vertex is the same distance from the line of reflection, but on the opposite side.

Each vertex of the original triangle is the same distance from the y-axis as its reflected counterpart: $(1, 1)$ is 1 unit right of the y-axis, and $(-1, 1)$ is 1 unit left. The dashed purple lines show these matching distances.

Rotations (Turns)

A rotation turns a figure around a fixed point (the center of rotation) by a specified angle. When the center is the origin, the rules are clean and predictable.

90 degrees counterclockwise about the origin:

$(x, y) \to (-y, x)$

180 degrees about the origin:

$(x, y) \to (-x, -y)$

270 degrees counterclockwise (same as 90 degrees clockwise) about the origin:

$(x, y) \to (y, -x)$

A rotation preserves size and shape. The figure ends up in a different position and orientation, but all side lengths and angles remain unchanged.

Example 3: Rotate a point 90 degrees counterclockwise

Rotate the point $(2, 5)$ by $90\degree$ counterclockwise about the origin.

Apply the rule $(x, y) \to (-y, x)$:

$(2, 5) \to (-5, 2)$

Answer: The image is $(-5, 2)$.

To verify: the distance from the origin stays the same. Original distance: $\sqrt{2^2 + 5^2} = \sqrt{29}$. New distance: $\sqrt{(-5)^2 + 2^2} = \sqrt{29}$. Confirmed.

Dilations (Resizing)

A dilation scales a figure by a factor $k$ from a center point. When the center is the origin, the rule is:

$(x, y) \to (kx, ky)$

• If $k > 1$, the figure gets larger (an enlargement)
• If $0 < k < 1$, the figure gets smaller (a reduction)
• If $k = 1$, nothing changes

Dilations produce similar figures — same shape, same angles, but different size. Unlike the other three transformations, a dilation is not a rigid motion because it changes the dimensions of the figure.

For example, if you dilate a triangle by a factor of 2, every side length doubles, but all three angles stay the same.

Example 4: Dilate a triangle by factor 1/2

Dilate the triangle with vertices $(2, 4)$, $(6, 4)$, $(4, 8)$ by a scale factor of $\frac{1}{2}$ centered at the origin.

Apply $(x, y) \to \left(\frac{1}{2}x,\ \frac{1}{2}y\right)$:

$(2, 4) \to (1, 2)$

$(6, 4) \to (3, 2)$

$(4, 8) \to (2, 4)$

Answer: The dilated triangle has vertices $(1, 2)$, $(3, 2)$, $(2, 4)$. Each coordinate is half the original, so every side length is halved while the shape remains similar.

Rigid vs. Non-Rigid Transformations

Transformations fall into two categories based on whether they preserve size:

A rigid transformation (or isometry) preserves both side lengths and angles. The original and its image are congruent — identical in every measurement.

A dilation preserves angles but changes side lengths by the scale factor. The original and its image are similar — same shape, different size.

This distinction matters: if a problem asks whether two figures are congruent, only rigid transformations can establish that. If it asks whether they are similar, a dilation (possibly combined with rigid motions) can connect them.

Line Symmetry

A figure has line symmetry (also called reflective symmetry) if you can draw a line through it so that one half is the mirror image of the other. This line is called the line of symmetry or axis of symmetry.

Common examples:

The key pattern: a regular polygon with $n$ sides has exactly $n$ lines of symmetry.

Example 5: Lines of symmetry in a regular hexagon

How many lines of symmetry does a regular hexagon have?

A regular hexagon has 6 equal sides and 6 equal angles. Each line of symmetry either connects two opposite vertices or connects the midpoints of two opposite sides.

• Vertex-to-vertex lines: 3 (connecting each pair of opposite vertices)
• Midpoint-to-midpoint lines: 3 (connecting the midpoints of each pair of opposite sides)

$\text{Total lines of symmetry} = 3 + 3 = 6$

Answer: A regular hexagon has 6 lines of symmetry.

Rotational Symmetry

A figure has rotational symmetry if it looks the same after rotating less than $360\degree$ about its center. The order of rotational symmetry is the number of positions in which the figure looks identical during a full $360\degree$ rotation.

To find the angle of rotation, divide $360\degree$ by the order:

$\text{angle of rotation} = \frac{360\degree}{n}$

where $n$ is the order.

Examples:

A square has order 4 because it looks the same at $90\degree$, $180\degree$, $270\degree$, and $360\degree$ — four positions total.

A figure with no rotational symmetry (like a scalene triangle) has order 1 — it only looks the same after a full $360\degree$ rotation.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Translations slide every point by the same amount: $(x, y) \to (x + a,\ y + b)$
• Reflections flip a figure across a line — over the x-axis: $(x, y) \to (x, -y)$; over the y-axis: $(x, y) \to (-x, y)$; over $y = x$: $(x, y) \to (y, x)$
• Rotations turn a figure about a point — $90\degree$ CCW: $(x, y) \to (-y, x)$; $180\degree$: $(x, y) \to (-x, -y)$; $270\degree$ CCW: $(x, y) \to (y, -x)$
• Dilations scale by factor $k$: $(x, y) \to (kx, ky)$ — enlargement if $k > 1$, reduction if $0 < k < 1$
• Translations, reflections, and rotations are rigid (preserve size) — dilations are non-rigid (change size but preserve shape)
• A regular polygon with $n$ sides has $n$ lines of symmetry and rotational symmetry of order $n$

Return to Geometry for more topics in this section.