Translations

Transformations of Figures

Today we will discuss translation, reflections, and rotations.

Multiple Choice

What is a transformation?

• one-to-one mapping of points in a plan such that each point in the pre-image has a unique image and each point in the image has a pre-image

• a rigid transformation moving all points in a geometric figure the same distance and same direction

• a rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line

• a rigid transformation where each point on the figure is rotated about a given point

Multiple Select

A rigid or isometric transformation preserves/does not change the ____ and ____ of the pre-image.

• size

• shape

• orientation

Multiple Choice

What is a translation?

• one-to-one mapping of points in a plan such that each point in the pre-image has a unique image and each point in the image has a pre-image

• a rigid transformation moving all points in a geometric figure the same distance and same direction

• a rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line

• a rigid transformation where each point on the figure is rotated about a given point

Multiple Choice

What is a reflection?

• one-to-one mapping of points in a plan such that each point in the pre-image has a unique image and each point in the image has a pre-image

• a rigid transformation moving all points in a geometric figure the same distance and same direction

• a rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line

• a rigid transformation where each point on the figure is rotated about a given point

Multiple Choice

What is a rotation?

• one-to-one mapping of points in a plan such that each point in the pre-image has a unique image and each point in the image has a pre-image

• a rigid transformation moving all points in a geometric figure the same distance and same direction

• a rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line

• a rigid transformation where each point on the figure is rotated about a given point

Translations

• A rigid transformation moving all points in a geometric figure the same distance and the same direction.

• Represented in coordinate notation:

   $\left(x,y\right)\rightarrow\left(x+a,y+b\right)$  where  $a$  is a horizontal shift and  $b$  causes a vertical shift.

Multiple Choice

Parallelogram  $ABCD$  was transformed to form parallelogram  $A'B'C'D'$  .

Which rule describes the transformation that was used to form parallelogram  $A'B'C'D'$  ?

•  $\left(-x,y-3\right)$  

•  $\left(x+10,y-3\right)$  

•  $\left(x-10,y-3\right)$  

•  $\left(x+10,y+3\right)$  

Reflection

• A rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line (a line of symmetry).

• Reflection Across the x-axis: $\left(x,y\right)\rightarrow\left(x,-y\right)$  

• Reflection Across the y-axis  $\left(x,y\right)\rightarrow\left(-x,y\right)$  

• Reflection Across the y = x  $\left(x,y\right)\rightarrow\left(y,x\right)$  

• Reflection Across the y = -x  $\left(x,y\right)\rightarrow\left(-y,-x\right)$  

Multiple Choice

Which rule describes a reflection across the x-axis?

• $\left(x,y\right)\rightarrow\left(x,-y\right)$

• $\left(x,y\right)\rightarrow\left(-x,y\right)$

• $\left(x,y\right)\rightarrow\left(y,x\right)$

• $\left(x,y\right)\rightarrow\left(-y,-x\right)$

Multiple Choice

Which rule describes a reflection across the y-axis?

• $\left(x,y\right)\rightarrow\left(x,-y\right)$

• $\left(x,y\right)\rightarrow\left(-x,y\right)$

• $\left(x,y\right)\rightarrow\left(y,x\right)$

• $\left(x,y\right)\rightarrow\left(-y,-x\right)$

Multiple Choice

Which rule describes a reflection across the y = x?

• $\left(x,y\right)\rightarrow\left(x,-y\right)$

• $\left(x,y\right)\rightarrow\left(-x,y\right)$

• $\left(x,y\right)\rightarrow\left(y,x\right)$

• $\left(x,y\right)\rightarrow\left(-y,-x\right)$

Multiple Choice

Which rule describes a reflection across the y = -x?

• $\left(x,y\right)\rightarrow\left(x,-y\right)$

• $\left(x,y\right)\rightarrow\left(-x,y\right)$

• $\left(x,y\right)\rightarrow\left(y,x\right)$

• $\left(x,y\right)\rightarrow\left(-y,-x\right)$

Rotations

• A rigid transformation where each point on the figure is rotated about a given point

• Rotation of 90° counterclockwise or 270° clockwise about the origin $\left(x,y\right)\rightarrow\left(-y,x\right)$  

• Rotation of 180° counterclockwise or 180° clockwise about the origin  $\left(x,y\right)\rightarrow\left(-x,-y\right)$  

• Rotation of 270° counterclockwise or 90° clockwise about the origin  $\left(x,y\right)\rightarrow\left(y,-x\right)$  

Multiple Choice

Which rule describes a rotation 90° counterclockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Multiple Choice

Which rule describes a rotation 270° clockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Multiple Choice

Which rule describes a rotation 180° counterclockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Multiple Choice

Which rule describes a rotation 180° clockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Multiple Choice

Which rule describes a rotation 270° counterclockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Multiple Choice

Which rule describes a rotation 90° clockwise about the origin?

• $\left(x,y\right)\rightarrow\left(-y,x\right)$

• $\left(x,y\right)\rightarrow\left(-x,-y\right)$

• $\left(x,y\right)\rightarrow\left(y,-x\right)$

Non-Rigid or Non-Isometric Transformations

A transformation that preserves similarity and reductions and enlargements that do not preserve similarity.

Dilation

• a similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation

•  $\left(x,y\right)\rightarrow\left(kx,ky\right)$  where k is a scale factor when the center of dilation is the origin

•  $\left(x,y\right)\rightarrow\left(k\left(x-a\right)+a,k\left(y-b\right)+b\right)$  where k is a scale factor and (a,b) is the center of dilation

Multiple Choice

Which rule can be used to describe a dilation when the center of dilation is the origin?

• $\left(x,y\right)\rightarrow\left(kx,ky\right)$

• $\left(x,y\right)\rightarrow\left(k\left(x-a\right)+a,k\left(y-b\right)+b\right)$

Multiple Choice

Which rule can be used to describe a dilation when the center of dilation is the not origin (represented by (a,b))?

• $\left(x,y\right)\rightarrow\left(kx,ky\right)$

• $\left(x,y\right)\rightarrow\left(k\left(x-a\right)+a,k\left(y-b\right)+b\right)$

Multiple Select

What are examples of rigid transformations?

• Translations

• Reflections

• Rotations

• Dilations

Multiple Select

What are examples of non-rigid transformations?

• Translations

• Reflections

• Rotations

• Dilations