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# Composition of Transformations

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Composite Transformations

Look at the following diagram. It involves two translations. Identify the two translations of triangle $ABC$ .

### Watch This

First watch this video to learn about composite transformations.

Then watch this video to see some examples.

### Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A composite transformation is when two or more transformations are performed on a figure (called the preimage) to produce a new figure (called the image).

#### Example A

Describe the transformations in the diagram below. The transformations involve a reflection and a rotation.

Solution:  First line $AB$ is rotated about the origin by $90^\circ$ CCW.

Then the line $A^\prime B^\prime$ is reflected about the $y$ -axis to produce line $A^{\prime \prime}B^{\prime \prime}$ .

#### Example B

Describe the transformations in the diagram below.

Solution: The flag in diagram S is rotated about the origin $180^\circ$ to produce flag T. You know this because if you look at one point you notice that both $x$ - and $y$ -coordinate points is multiplied by -1 which is consistent with a $180^\circ$ rotation about the origin. Flag T is then reflected about the line $x = -8$ to produce Flag U.

#### Example C

Triangle $ABC$ where the vertices of $\Delta ABC$ are $A(-1, -3)$ , $B(-4, -1)$ , and $C(-6, -4)$ undergoes a composition of transformations described as:

a) a translation 10 units to the right, then

b) a reflection in the $x$ -axis.

Draw the diagram to represent this composition of transformations. What are the vertices of the triangle after both transformations are applied?

Solution:

Triangle $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$ is the final triangle after all transformations are applied. It has vertices of $A^{\prime \prime}(9, 3)$ , $B^{\prime \prime}(6, 1)$ , and $C^{\prime \prime}(4, 4)$ .

#### Concept Problem Revisited

$\Delta ABC$ moves over 6 to the left and down 5 to produce $\Delta A^\prime B^\prime C^\prime$ . Then $\Delta A^\prime B^\prime C^\prime$ moves over 14 to the right and up 3 to produce $\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$ . These translations are represented by the blue arrows in the diagram.

All together $\Delta ABC$ moves over 8 to the right and down 2 to produce $\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$ . The total translations for this movement are seen by the green arrow in the diagram above.

### Vocabulary

Image
In a transformation, the final figure is called the image .
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Dilation
A dilation is a transformation that enlarges or reduces the size of a figure.
Translation
A translation is an example of a transformation that moves each point of a shape the same distance and in the same direction. Translations are also known as slides .
Rotation
A rotation is a transformation that rotates (turns) an image a certain amount about a certain point.
Reflection
A reflection is an example of a transformation that flips each point of a shape over the same line.
Composite Transformation
A composite transformation is when two or more transformations are combined to form a new image from the preimage.

### Guided Practice

1. Describe the transformations in the diagram below. The transformations involve a rotation and a reflection.

2. Triangle $XYZ$ has coordinates $X (1, 2)$ , $Y (-3, 6)$ and $Z (4, 5)$ .The triangle undergoes a rotation of 2 units to the right and 1 unit down to form triangle $X^\prime Y^\prime Z^\prime$ . Triangle $X^\prime Y^\prime Z^\prime$ is then reflected about the $y$ -axis to form triangle $X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}$ . Draw the diagram of this composite transformation and determine the vertices for triangle $X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}$ .

3. The coordinates of the vertices of $\Delta JAK$ are $J(1, 6)$ , $B(2, 9)$ , and $C(7, 10)$ .

a) Draw and label $\Delta JAK$ .

b) $\Delta JAK$ is reflected over the line $y=x$ . Graph and state the coordinates of $\Delta J^\prime A^\prime K^\prime$ .

c) $\Delta J^\prime A^\prime K^\prime$ is then reflected about the $x$ -axis. Graph and state the coordinates of $\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}$ .

d) $\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}$ undergoes a translation of 5 units to the left and 3 units up. Graph and state the coordinates of $\Delta J^{\prime \prime \prime} A^{\prime \prime \prime} K^{\prime \prime \prime}$ .

1. The transformations involve a reflection and a rotation. First line $AB$ is reflected about the $y$ -axis to produce line $A^\prime B^\prime$ .

Then the line $A^\prime B^\prime$ is rotated about the origin by $90^\circ$ CCW to produce line $A^{\prime \prime} B^{\prime \prime}$ .

2.

3.

### Practice

1. A point $X$ has coordinates (-1, -8). The point is reflected across the $y$ -axis to form $X^\prime$ . $X^\prime$ is translated over 4 to the right and up 6 to form $X^{\prime \prime}$ . What are the coordinates of $X^\prime$ and $X^{\prime \prime}$ ?
2. A point $A$ has coordinates (2, -3). The point is translated over 3 to the left and up 5 to form $A^\prime$ . $A^\prime$ is reflected across the $x$ -axis to form $A^{\prime \prime}$ . What are the coordinates of $A^\prime$ and $A^{\prime \prime}$ ?
3. A point $P$ has coordinates (5, -6). The point is reflected across the line $y = -x$ to form $P^\prime$ . $P^\prime$ is rotated about the origin $90^\circ$ CW to form $P^{\prime \prime}$ . What are the coordinates of $P^\prime$ and $P^{\prime \prime}$ ?
4. Line $JT$ has coordinates $J(-2, -5)$ and $T(2, 3)$ . The segment is rotated about the origin $180^\circ$ to form $J^\prime T^\prime$ . $J^\prime T^\prime$ is translated over 6 to the right and down 3 to form $J^{\prime \prime} T^{\prime \prime}$ . What are the coordinates of $J^\prime T^\prime$ and $J^{\prime \prime} T^{\prime \prime}$ ?
5. Line $SK$ has coordinates $S(-1, -8)$ and $K(1, 2)$ . The segment is translated over 3 to the right and up 3 to form $S^\prime K^\prime$ . $S^\prime K^\prime$ is rotated about the origin $90^\circ$ CCW to form $S^{\prime \prime} K^{\prime \prime}$ . What are the coordinates of $S^\prime K^\prime$ and $S^{\prime \prime} K^{\prime \prime}$ ?
6. A point $K$ has coordinates (-1, 4). The point is reflected across the line $y=x$ to form $K^\prime$ . $K^\prime$ is rotated about the origin $270^\circ$ CW to form $K^{\prime \prime}$ . What are the coordinates of $K^\prime$ and $K^{\prime \prime}$ ?

Describe the following composite transformations:

1. Explore what happens when you reflect a shape twice, over a pair of parallel lines. What one transformation could have been performed to achieve the same result?
2. Explore what happens when you reflect a shape twice, over a pair of intersecting lines. What one transformation could have been performed to achieve the same result?
3. Explore what happens when you reflect a shape over the x-axis and then the y-axis. What one transformation could have been performed to achieve the same result?
4. A composition of a reflection and a translation is often called a glide reflection. Make up an example of a glide reflection. Why do you think it's called a glide reflection?

### Vocabulary Language: English

Reflection

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.
Rotation

Rotation

A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.
Glide Reflection

Glide Reflection

A reflection followed by a translation where the line of reflection is parallel to the direction of translation is called a glide reflection or a walk.
Composite Transformation

Composite Transformation

A composite transformation, also known as composition of transformation, is a series of multiple transformations performed one after the other.