Geometric Transformations: Translations, Rotations, and Reflections

Geometric transformations are operations that change the position, size, or orientation of geometric figures while preserving certain properties. In this guide, we'll explore three fundamental transformations:

Translations: Move figures in a straight line without changing their size or shape.
Rotations: Turn figures around a fixed point without changing their size or shape.
Reflections: Flip figures across a line or point, creating a mirror image.

1. Translations

What is a Translation?

A translation moves every point of a figure the same distance in the same direction. It's like sliding the figure without rotating or flipping it.

Translation Formula:

If a point (x, y) is translated by (a, b), the new coordinates are:

(x, y) → (x + a, y + b)

Where:

a = horizontal shift (positive for right, negative for left)
b = vertical shift (positive for up, negative for down)

Examples of Translations

Example 1: Basic Translation

Translate the point (3, 2) by 4 units right and 5 units up.

Solution:

Using the formula (x, y) → (x + a, y + b)

(3, 2) → (3 + 4, 2 + 5) = (7, 7)

The translated point is (7, 7).

Example 2: Translating a Triangle

Translate the triangle with vertices at (0, 0), (2, 0), and (1, 2) by 3 units left and 1 unit down.

Solution:

Here a = -3 (left) and b = -1 (down)

Vertex 1: (0, 0) → (0 + (-3), 0 + (-1)) = (-3, -1)

Vertex 2: (2, 0) → (2 + (-3), 0 + (-1)) = (-1, -1)

Vertex 3: (1, 2) → (1 + (-3), 2 + (-1)) = (-2, 1)

The translated triangle has vertices at (-3, -1), (-1, -1), and (-2, 1).

Interactive Translation Demo

Adjust the sliders to translate the blue square:

Horizontal shift (a): 0

Vertical shift (b): 0

Original: (0, 0)
Translated: (0, 0)

Practice Problems

Problem 1: Translate the point (5, -3) by 2 units left and 4 units up.

Problem 2: Rectangle ABCD has vertices at A(1, 1), B(5, 1), C(5, 3), and D(1, 3). Translate it by the vector ⟨-3, 2⟩ and find the new vertices.

Real-World Applications

Translations are used in:

Animation and game development (moving characters)
Computer graphics (positioning objects)
Engineering designs (adjusting component positions)
Navigation systems (GPS coordinate adjustments)

Angle	cos(θ)	sin(θ)	Simplified Formula
90°	0	1	(x, y) → (-y, x)
180°	-1	0	(x, y) → (-x, -y)
270°	0	-1	(x, y) → (y, -x)
360°	1	0	(x, y) → (x, y)

Line of Reflection	Formula
x-axis	(x, y) → (x, -y)
y-axis	(x, y) → (-x, y)
y = x (diagonal)	(x, y) → (y, x)
y = -x (other diagonal)	(x, y) → (-y, -x)
Line through origin with slope m	Complex formula that depends on the angle of the line. For a line with angle θ, first rotate coordinates by -θ, reflect across x-axis, then rotate back by θ.
Origin (point reflection)	(x, y) → (-x, -y)
Point (h, k)	(x, y) → (2h-x, 2k-y)

Summary of Transformations

Transformation	Formula	Key Properties
Translation	(x, y) → (x + a, y + b)	Preserves orientation, shape, size, and angles Moves all points the same distance in the same direction
Rotation (origin)	(x, y) → (x·cos(θ) - y·sin(θ), x·sin(θ) + y·cos(θ))	Preserves shape, size, and distances from center Changes orientation for rotations not equal to 360°
Reflection	Depends on line of reflection (See formulas in the Reflections section)	Preserves shape and size Reverses orientation (creates mirror image)

Important Concepts to Remember

Transformations can be combined in sequence (compositions).
The order of transformations matters - different orders can produce different results.
Translations, rotations, and reflections are all examples of rigid transformations (isometries) that preserve the shape and size of objects.
Other transformations not covered here include dilations (scaling) and shears.