## 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

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