## Key Concepts

- Perform Congruence Transformations
- Identify transformations

## Perform Congruence Transformations

### What is a transformation?

A transformation is an operation that changes a geometric figure to produce a new figure.

### Types of transformations:

Transformations can be divided into 4 types based on the image.

**Translation****Rotation****Reflection****Dilation**

#### 1. Translation:

A translation moves every point of a figure in the same direction and for the same distance.

#### 2. Rotation:

A rotation turns the figure to its fixed point called as the center of rotation.

#### 3. Reflection:

A reflection uses a line of reflection to create a mirror image of the original figure.

#### 4. Dilation:

Dilation is the process of increasing or decreasing the size of the image without changing its shape.

### What is a congruence transformation?

A transformation that changes the position of the figure without changing its size or shape is called a congruence transformation.

Translations, reflections, and rotations are the three types of congruence transformations.

### Identify transformations

**Example 1:**

Identify the type of transformation in the following images.

**Solution:**

a. Reflection in a horizontal line.

b.Rotation about a point.

c.Translation in a straight path.

**Example 2:**

The vertices of a quadrilateral ABCD are A (– 4, 3), B (– 2, 4), C (– 1, 1), and D (– 3, 1). Draw a figure and its image after the translation (x,y)→(x+5,y−2).x,y→x+5,y−2.

**Solution:**

Draw a quadrilateral ABCD. Find the translation of each vertex by adding 5 to its x – coordinate and subtract 2 from its y – coordinate.

**Example 3:**

In the given figure, use a reflection in X–axis to draw the other half of the pattern.

**Solution:**

To find the corresponding vertex in the image, multiply the Y–coordinate of the vertex by – 1.

**Example 4:**

Graph AB and CD with the given vertices, A (– 3, 1), B (– 1, 3), C (1, 3), D (3, 1). Give the angle and direction of rotation.

**Solution: **

We first draw AB and CD with the given vertices.

From the above figure,

*m*∠*AOC*=*m*∠*BOD*=90°

∴The given vertices form an angle of 90°clockwise rotation.

**Example 5:**

The vertices of a triangle ABC are A (4, 4), B (6, 6), C (7, 4), and the notation (x,y)→(x+1, y−3)(x,y)→(x+1, y−3) indicates the translation of ∆ABC to ∆DEF. Prove that ∆ABC≅∆DEF to verify the translation, a congruence transformation.

**Solution:**

From the above figure and the given vertices,

## Exercise

- Identify the transformation in the given figure.

- In the given image ABCD with the coordinates Draw its image after the translation.

- Describe the translation with the given coordinates: 2 units to the right, 1 unit down.
- Draw the other half of the given figure using –axis reflection.

- Graph AB and CD using the coordinates A (1, 2), B (3, 0), C (2, –1), D (2,
**–**3) and give the angle of rotation. - Find the corresponding point in the figure with the given point and translation of an image.

Point on image: (4, 0); translation: - Describe the translation with the given coordinates: 6 units to the right, 3 units down.
- Identify the transformation in the given figure

- Draw the other half of the given figure using –axis reflection.

- Identify the type of transformation in the given image.

### Concept Map

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: