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Congruence Transformations

- Calculus
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Suppose you are presented with two triangles and are asked to say what you notice about them. You notice that both triangles appear to be identical, and one is just oriented slightly differently. You may know that the word which describes this situation is **congruency**; as in, the triangles are **congruent** to one another.

You may also be asked to determine what change took place on the triangle which is oriented differently. In this case, it was a congruence transformation because the triangles are identical. In this article, we will discuss** congruence transformations** in detail.

Before discussing congruence transformations, let's first consider the concepts and definitions of congruence and transformations separately. What does it mean when objects or shapes are congruent?

Two objects are said to be **congruent** if they have the same shape and dimensions (size). For example, an object and its reflection in a mirror are congruent, whereas an object and its photograph are not congruent, as the photograph is a scaled-down representation of the object.

Two shapes are congruent if one of them can be moved (without changing its shape or size) such that it fits** exactly **over the other. Now, what is a transformation, as performed on a shape or object?

A **transformation** in Geometry is defined as the change in the relative position or size of an object.

Next, let's define congruence transformations, which is a combination of these two concepts:

A **congruence transformation** is the movement or repositioning of a shape such that it produces a shape which is congruent to the original.

Note that not all transformations are congruence transformations. For example, a transformation which changes the object's size is not a congruence transformation. This is because if two objects do not overlap exactly after the application of a transformation, the objects are incongruent, and the transformation was not a congruence transformation.

Congruence transformations are useful because they allow us to **prove** congruency between shapes. Depending on the case, we can prove the congruency of shapes using one congruence transformation or **a sequence of multiple** congruence transformations.

There are three types of congruence transformations:

Translation (up, down, left, right), such as sliding a piece of paper on a tabletop

Reflection (mirror image of an object)

Rotation (about a point on the object or an externally located object)

We will now discuss each of these types of transformations in more detail along with some examples.

Let's explore in more detail each of the three different congruence transformations:

- Translations
- Reflections
- Rotations

A translation is where we take a shape and move it somewhere else. The actual shape doesn't change, but instead it is just **"copied and pasted"** somewhere else. The examples below illustrate this concept.

**Describe the transformation that maps triangle ABC onto A'B'C'. **

**Solution**

From the figure, we can see that triangle ABC has simply been moved to another location, resulting in triangle A'B'C'. When describing translations, we use vectors. In the above diagram, if we look at point A, we can see it has been moved three spaces to the right and four spaces up to get to the point A'. This is the same for point B and C. Thus, we can say that the shape has been translated by the vector .

Using **vectors** enables us to describe translations accurately. Recall that vectors are denoted as , where a represents the units it has been moved horizontally and b represents the units it has moved vertically. If we have a negative value for a, we move the shape to the left. If we have a negative value for b, we move the shape down.

**Translate the below quadrilateral by the vector .**

**Solution**

To translate the shape, it helps to look at each point individually. If we take point A, we need to move it 4 spaces to the left and 3 spaces down to get to the point A'. If we do the same for points B, C, and D, we get the below quadrilateral A'B'C'D'.

A reflection is essentially a mirror image of a shape reflected across a particular line. In order to reflect a shape, we need to know the equation of the line we are reflecting over. We can also reflect shapes across either the x- or y-axis. Here, we look at some examples of reflections.

**Reflect the shape ABCD across the y-axis.**

Reflection Example 1 - StudySmarter Originals

**Solution**

Since we are reflecting across the y-axis, the shape will only move horizontally and only the x-values will change. To reflect across the y-axis, we multiply the x-values by -1, like so:

- Point A (2,5) is reflected to point A' (-2,5)
- Point B (4,5) is reflected to point B' (-4,5)
- Point C (4,3) is reflected to point C' (-4,3)
- Point D (2,3) is reflected to point D' (-2,3)

See the reflected points A', B', C', and D' plotted in the graph below.

**In the shape below, describe the single transformation that maps the triangle ABC to triangle A'B'C'. **

**Solution**

If we look closely at the figure above, we can see that the shape ABC has been reflected over the line . Therefore, we have a reflection over the line .

This line has a positive slope of 1, and it passes directly through the origin (0,0). To reflect over the line y=x, we swap the points' x- and y-coordinates, like so:

- Point A (3,-1) is reflected to point A' (-1,3)
- Point B (3,-3) is reflected to point B' (-3,3)
- Point C (1,-3) is reflected to point C' (-3,1)

A rotation is when a shape is rotated about a particular point. Common rotations are 90 degrees and 180 degrees clockwise or counterclockwise. When rotating a shape, we must take note of the center of rotation.

**Describe the single transformation that maps shape ABCDE onto A'B'C'D'E'. **

**Solution**

We can tell by looking at the transformed shape that there has been a rotation of 90 degrees counterclockwise. However, we need to determine the center of rotation. If we look at the points E and E', we can see that a right angle is formed about the origin. Therefore, the origin is the center of rotation for this shape's congruence transformation. Thus, we have a rotation of 90 degrees counterclockwise about the origin.

To rotate an object 90 degrees about the origin, we perform the following adjustment to our coordinates:

(x,y) becomes (-y,x), like so:

- Point A (2,3) is rotated to point A' (-3,2)
- Point B (4,3) is rotated to point B' (-3,4)
- Point C (4,0) is rotated to point C' (0,4)
- Point D (3,1) is rotated to point D' (-1,3)
- Point E (2,0) is rotated to point E' (0,2)

**Rotate the triangle ABC 180 degrees about the origin. **

**Solution**

If we consider point A's distance from the origin, we can see that it is 1 unit to the right and 3 units up. If we rotate it 180 degrees clockwise, it will be 1 unit to the left of the origin and three units down. The same applies to points B and C. We have the below rotation.

To rotate an object 180 degrees about the origin, we perform the following adjustment to our coordinates:

(x,y) becomes (-x,-y), like so:

- Point A (1,3) is rotated to point A' (-1,-3)
- Point B (3,1) is rotated to point B' (-3,-1)
- Point C (4,4) is rotated to point C' (-4,-4)

We will now have a look at two important theorems on congruence transformations. These are the reflection in parallel lines theorem and the reflections in intersecting lines theorem, and they help us to identify congruence transformations. Let's start by talking about the reflections in parallel lines theorem.

The reflection in parallel lines theorem states that if we reflect a shape over two parallel lines, first about line A and then over line B, the resultant triangle is the same as translating the original triangle. Additionally, the orientation of the resulting translation is perpendicular to the parallel lines, and the magnitude will always be two times the distance between the parallel lines.

The reflections in intersecting lines theorem state that, if we reflect a shape twice over two intersecting lines, the resultant shape can also be obtained by rotation of the shape about the point of intersection of the lines. The angle of rotation is always twice the angle between the intersecting lines.

There are three types of congruent transformations: reflection, translation, and rotations.

Congruence transformations do not change the shape or size of the object.

If one shape can be obtained from another by a sequence of congruence transformations, the shapes are congruent.

Congruency between two shapes can be proved by a single congruence transformation or by a sequence of congruence transformations.

The dilation transformation does not preserve congruence as it changes the size of the shape.

The 3 types of congruence transformations are:

Rotation

Reflection

Translation

A congruence transformation should not change the shape and size of the object when applied to the object.

If one object can be moved to get the other using a sequence of congruence transformations then the objects are congruent.

More about Congruence Transformations

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