Introduction to Transformations - Geometry | Luna Learn 24

Introduction to Transformations

This lesson provides an overview of the four fundamental types of geometric transformations: translations, rotations, reflections, and dilations. Understanding these transformations is crucial in Geometry as they form the basis for many concepts and applications.

Outline

Definition and Overview
Translations
Rotations
Reflections
Dilations
Examples and Diagrams
Exercises with Solutions
Homework and Revision Section

1. Definition and Overview

Transformations are operations that move or change a shape in some way while preserving its fundamental properties. The main types of transformations are translations, rotations, reflections, and dilations.

2. Translations

A translation moves every point of a shape the same distance in the same direction. It slides the shape without rotating or flipping it.

Example 1: Translation

Translate triangle \( \triangle ABC \) 5 units to the right and 3 units up. If \( A(1, 2) \), \( B(3, 4) \), and \( C(5, 2) \), what are the new coordinates?

New coordinates:

A'(6, 5)
B'(8, 7)
C'(10, 5)

3. Rotations

A rotation turns a shape around a fixed point called the center of rotation. The angle of rotation determines how far the shape is turned.

Example 2: Rotation

Rotate rectangle \( ABCD \) \( 90^\circ \) clockwise around the origin. If \( A(2, 1) \), \( B(5, 1) \), \( C(5, 4) \), and \( D(2, 4) \), what are the new coordinates?

New coordinates:

A'(-1, 2)
B'(-1, 5)
C'(-4, 5)
D'(-4, 2)

4. Reflections

A reflection flips a shape over a line, creating a mirror image. The line of reflection is the axis over which the shape is flipped.

Example 3: Reflection

Reflect triangle \( \triangle PQR \) over the y-axis. If \( P(3, 2) \), \( Q(5, 4) \), and \( R(4, 1) \), what are the new coordinates?

New coordinates:

P(-3, 2)
Q(-5, 4)
R(-4, 1)

5. Dilations

A dilation changes the size of a shape but keeps its shape and angles the same. It enlarges or reduces the shape based on a scale factor and a center of dilation.

Example 4: Dilation

Dilate triangle \( \triangle XYZ \) by a scale factor of 2. If \( X(1, 1) \), \( Y(3, 1) \), and \( Z(2, 4) \), what are the new coordinates?

New coordinates:

X(2, 2)
Y(6, 2)
Z(4, 8)

Examples and Practice

Example 5: Combining Transformations

Triangle \( \triangle ABC \) is translated 3 units left and 2 units down, then reflected over the x-axis. If \( A(4, 5) \), \( B(7, 5) \), and \( C(6, 3) \), what are the final coordinates?

After translation:

A(1, 3)
B(4, 3)
C(3, 1)

After reflection:

A(1, -3)
B(4, -3)
C(3, -1)

Example 6: Transformation on a Coordinate Grid

A square with vertices \( (2, 2) \), \( (2, 5) \), \( (5, 5) \), and \( (5, 2) \) is rotated \( 90^\circ \) counterclockwise around the origin. What are the new coordinates of the vertices?

New coordinates:

(2, 2) → (-2, 2)
(2, 5) → (-5, 2)
(5, 5) → (-5, -2)
(5, 2) → (-2, -5)

Homework

Complete the following exercises:

Translate the given quadrilateral \( (1, 1) \), \( (1, 4) \), \( (4, 4) \), and \( (4, 1) \) by -2 units in the x direction and 3 units in the y direction.
Rotate the triangle \( \triangle DEF \) with vertices \( (3, 2) \), \( (4, 5) \), and \( (6, 2) \) \( 180^\circ \) around the origin.
Reflect the point \( (7, -2) \) over the line \( y = -x \).
Dilate the shape with vertices \( (1, 3) \), \( (2, 1) \), and \( (3, 2) \) by a scale factor of 0.5 centered at the origin.