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.
![What is a transformation?](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1522.png)
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.
![Translation](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1523.png)
2. Rotation:
A rotation turns the figure to its fixed point called as the center of rotation.
![Rotation:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1524.png)
3. Reflection:
A reflection uses a line of reflection to create a mirror image of the original figure.
![Reflection](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1525.png)
4. Dilation:
Dilation is the process of increasing or decreasing the size of the image without changing its shape.
![Dilation:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1526.png)
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.
![What is a congruence transformation?](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1540.png)
Identify transformations
Example 1:
Identify the type of transformation in the following images.
![Example 1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1527.png)
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.
![Solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1541.png)
![Solution table](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1542.png)
Example 3:
In the given figure, use a reflection in X–axis to draw the other half of the pattern.
![Example 3](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1528.png)
Solution:
To find the corresponding vertex in the image, multiply the Y–coordinate of the vertex by – 1.
![Solution table](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1529.png)
![Solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1530.png)
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.
![Example 4:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1531-1024x580.png)
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:
![Solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1532-1024x870.png)
From the above figure and the given vertices,
![From the above figure and the given vertices](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2359.png)
Exercise
- Identify the transformation in the given figure.
![Identify the transformation in the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1553.png)
- In the given image ABCD with the coordinates Draw its image after the translation.
![In the given image ABCD with the coordinates Draw its image after the translation.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1554.png)
- 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.
![other half of the given figure](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1555.png)
- 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
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1556.png)
- Draw the other half of the given figure using –axis reflection.
![Draw the other half of the given figure using –axis reflection.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1557.png)
- Identify the type of transformation in the given image.
Concept Map
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1537-1024x401.png)
![Congruence Transformations](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2022%2F09%2FCongruence-Transformations.png&w=1920&q=50)
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