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Composition of Transformations

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Composite Transformations
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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.

CK-12 Foundation Chapter10CompositeTransformationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10CompositeTransformationsB

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} .

Answers:

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?

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