Congruence Transformations

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

1. Identify the transformation in the given figure.

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

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

5. 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.
6. Find the corresponding point in the figure with the given point and translation of an image.
   Point on image: (4, 0); translation:
7. Describe the translation with the given coordinates: 6 units to the right, 3 units down.
8. Identify the transformation in the given figure

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

10. Identify the type of transformation in the given image.

Concept Map