Transformations — Translation, Rotation, Reflection & Dilation

Transformations in geometry are rules that send each point of a figure to a new point. The four main transformations students meet first are translation, rotation, reflection, and dilation.

The fast way to sort them is this: translation slides, rotation turns, reflection flips, and dilation resizes. Translation, rotation, and reflection preserve size and shape. Dilation preserves shape but changes size unless the scale factor is $1$.

What Each Type of Transformation Does

Translation moves every point the same distance in the same direction. Nothing turns or flips.

Rotation turns a figure around a fixed point called the center of rotation. The angle and direction matter.

Reflection flips a figure across a line, called the line of reflection. Points land the same perpendicular distance on the other side of that line.

Dilation enlarges or shrinks a figure from a fixed center by a scale factor. If $k > 1$, the image gets larger. If $0 < k < 1$, the image gets smaller.

Coordinate Rules You Can Use Safely

These shortcut rules work only under the stated condition. If the center or reflection line changes, the coordinate rule changes too.

For a translation by $(a, b)$,

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

For a rotation about the origin:

$$90^\circ \text{ counterclockwise}: (x, y) \to (-y, x)$$ $$180^\circ: (x, y) \to (-x, -y)$$ $$90^\circ \text{ clockwise}: (x, y) \to (y, -x)$$

For reflections:

$$\text{across the } x\text{-axis}: (x, y) \to (x, -y)$$ $$\text{across the } y\text{-axis}: (x, y) \to (-x, y)$$ $$\text{across the line } y = x: (x, y) \to (y, x)$$

For a dilation from the origin with scale factor $k$,

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

One Worked Example: The Same Point Under Four Transformations

Use the point $P(2, 1)$ so you can see exactly what changes each time.

If you translate it by $(3, -2)$, add those values to the coordinates:

$$(2, 1) \to (2 + 3, 1 - 2) = (5, -1)$$

If you rotate it $90^\circ$ counterclockwise about the origin, use $(x, y) \to (-y, x)$:

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

If you reflect it across the $y$-axis, change the sign of the $x$-coordinate:

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

If you dilate it from the origin with scale factor $2$, multiply both coordinates by $2$:

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

This single example shows the main difference quickly:

• translation changes position
• rotation changes orientation around a center
• reflection reverses the figure across a line
• dilation changes size by a scale factor

A polygon works the same way. Transform each vertex, then reconnect the image in the same order.

The Fastest Intuition: Ask What Stays Fixed

A transformation becomes easier to identify when you ask what stays fixed.

In a translation, the direction and distance stay fixed. In a rotation, the center stays fixed. In a reflection, the mirror line stays fixed. In a dilation, the center and scale factor stay fixed.

That is more reliable than memorizing rules alone. If you know the fixed reference, you can usually rebuild the correct process even when you forget a shortcut.

Common Mistakes With Transformations

Forgetting the condition on the rule

The shortcut $(x, y) \to (-y, x)$ works only for a $90^\circ$ counterclockwise rotation about the origin. Change the center, and you need a different process.

Mixing clockwise and counterclockwise

This is one of the most common errors in coordinate geometry. If the direction is not stated clearly, stop and identify it before you transform the point.

Assuming dilation preserves length

Dilation preserves shape, not actual size. Side lengths are multiplied by the scale factor, so the image is similar to the original, not usually congruent.

Reflecting across the wrong line

Reflecting across the $x$-axis, $y$-axis, and the line $y = x$ produces different results. The coordinates may look similar, but the rules are not interchangeable.

Where Transformations Are Used

Transformations appear in coordinate geometry, symmetry problems, graphing, computer graphics, map scaling, and basic modeling. They are useful whenever you need to describe how a shape moves, turns, flips, or changes size without redefining every point from scratch.

Try a Similar Problem

Try your own version with triangle $A(0, 1)$, $B(3, 1)$, and $C(1, 4)$. Translate it by $(2, -1)$, then reflect it across the $y$-axis, and check which transformation keeps the side lengths the same. If you want to go further, solve a similar problem with a rotation about the origin and compare the new coordinates.