Learn on PengiPengi Math (Grade 8)Chapter 6: Geometric Transformations and Similarity

Lesson 1: Introduction to Transformations and Translations

In this Grade 8 Pengi Math lesson from Chapter 6, students learn to define geometric transformations and distinguish between rigid and non-rigid transformations. The lesson focuses on translations as slides that preserve a figure's size and shape, with students applying coordinate rules of the form (x + a, y + b) to translate figures in the plane. Students also verify that translations preserve both distance and angle measures.

Section 1

What is a Transformation?

Property

A geometric transformation is a function that maps each point of a figure, called the pre-image, to a new point in a figure called the image. We denote the image of a point $A$ as $A'$ (read as "A prime").

Examples

Section 2

Introduction to Rigid Transformations

Property

A rigid transformation is a change in the position of a figure that preserves its size and shape. The three main types of rigid transformations are:

Translations (slides)
Reflections (flips)
Rotations (turns)

Examples

Section 3

Translations

Property

A translation is a rigid motion of the plane that moves horizontal lines to horizontal lines and vertical lines to vertical lines.

A translation preserves the lengths of line segments and the measures of angles.
For a translation, there is a pair $(a, b)$ , called the vector of the translation, such that the image of any point $(x, y)$ is the point $(x + a, y + b)$ .
Under a translation, the image of a line $L$ is a line $L'$ parallel to $L$ . Furthermore, translations take parallel lines to parallel lines. This is because a translation does not change the slope of a line.
A translation that does not leave every point fixed does not leave any point fixed.

Examples

The point $P(2, 5)$ is translated by the vector $(3, -4)$ . The image $P'$ has coordinates $(2+3, 5-4)$ , which is $(5, 1)$ .
A triangle with vertices $A(0,0)$ , $B(1,3)$ , and $C(4,1)$ is translated 2 units left and 5 units up. The new vertices are $A'(-2, 5)$ , $B'(-1, 8)$ , and $C'(2, 6)$ .
A translation moves point $Q(-1, 8)$ to $Q'(3, 5)$ . The vector for this translation is $(3 - (-1), 5 - 8)$ , which is $(4, -3)$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

What is a Transformation?

Property

Examples

Section 2

Introduction to Rigid Transformations

Property

A rigid transformation is a change in the position of a figure that preserves its size and shape. The three main types of rigid transformations are:

Translations (slides)
Reflections (flips)
Rotations (turns)

Examples

Section 3

Translations

Property

A translation is a rigid motion of the plane that moves horizontal lines to horizontal lines and vertical lines to vertical lines.

A translation preserves the lengths of line segments and the measures of angles.
For a translation, there is a pair $(a, b)$ , called the vector of the translation, such that the image of any point $(x, y)$ is the point $(x + a, y + b)$ .
Under a translation, the image of a line $L$ is a line $L'$ parallel to $L$ . Furthermore, translations take parallel lines to parallel lines. This is because a translation does not change the slope of a line.
A translation that does not leave every point fixed does not leave any point fixed.

Examples

The point $P(2, 5)$ is translated by the vector $(3, -4)$ . The image $P'$ has coordinates $(2+3, 5-4)$ , which is $(5, 1)$ .
A triangle with vertices $A(0,0)$ , $B(1,3)$ , and $C(4,1)$ is translated 2 units left and 5 units up. The new vertices are $A'(-2, 5)$ , $B'(-1, 8)$ , and $C'(2, 6)$ .
A translation moves point $Q(-1, 8)$ to $Q'(3, 5)$ . The vector for this translation is $(3 - (-1), 5 - 8)$ , which is $(4, -3)$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

What is a Transformation?

Property

Examples

Section 2

Introduction to Rigid Transformations

Property

A rigid transformation is a change in the position of a figure that preserves its size and shape. The three main types of rigid transformations are:

Translations (slides)
Reflections (flips)
Rotations (turns)

Examples

Section 3

Translations

Property

A translation is a rigid motion of the plane that moves horizontal lines to horizontal lines and vertical lines to vertical lines.

A translation preserves the lengths of line segments and the measures of angles.
For a translation, there is a pair $(a, b)$ , called the vector of the translation, such that the image of any point $(x, y)$ is the point $(x + a, y + b)$ .
Under a translation, the image of a line $L$ is a line $L'$ parallel to $L$ . Furthermore, translations take parallel lines to parallel lines. This is because a translation does not change the slope of a line.
A translation that does not leave every point fixed does not leave any point fixed.

Examples

The point $P(2, 5)$ is translated by the vector $(3, -4)$ . The image $P'$ has coordinates $(2+3, 5-4)$ , which is $(5, 1)$ .
A triangle with vertices $A(0,0)$ , $B(1,3)$ , and $C(4,1)$ is translated 2 units left and 5 units up. The new vertices are $A'(-2, 5)$ , $B'(-1, 8)$ , and $C'(2, 6)$ .
A translation moves point $Q(-1, 8)$ to $Q'(3, 5)$ . The vector for this translation is $(3 - (-1), 5 - 8)$ , which is $(4, -3)$ .