This document defines and provides examples of different geometric transformations including translations, reflections, rotations, and dilations. Translations involve sliding a figure along a vector. Reflections flip a figure across a line. Rotations turn a figure around a center point by a number of degrees. Dilations enlarge or shrink a figure proportionally away or towards a center point. Key terms like preimage, image, and scale factor are introduced and transformations are demonstrated on examples like triangles, quadrilaterals, and pentagons. Real-world connections are made to mirrors, clock hands, and the pupil of the eye.

1. Transformations
and More
Mejia, Jasper
Geometry, Per. 1
March 7, 2022
Translations
Reflections
Rotations
Dilations
Sponsored
by Desmos

2. But first… Key Terms!
- Transformation: a change in a geometric figures’ position, shape,
orientation, or size.
- Preimage: the figure before the transformation
- Image: the preimage after the transformation (labeled with a ‘)
- Isometry: a transformation in which the preimage and image are
congruent (the dimensions do not change)
- Rigid motion: a transformation that does not change the lengths
and angles of a figure.
- “→” means translated to. Example: Triangle ABC → Triangle A’B’C’

3. Translations - A translation is a slide and
isometry that maps all
points of a figure the same
distance in the same
direction.
- Trapezoid ACBD translated over to
trapezoid A’C’B’D’.
- The distance between the vertices of
the preimage and image are
congruent.
- AA’ = BB’ = CC’ = DD’
The position of the
basketball changes but
not the size.
Preimage: Trapezoid ACBD
Image: Trapezoid A’C’B’D’
Translation: (x, y) → (x + 8, y)

4. Translations - graphing
- The translation is labeled based on
how the vertices of the figure shifted.
- There can be a translation in both the
x and y directions
- Example: if you wanted to move the
figure seven spaces to the right, you
would add 7 to the x-coordinate of
each vertex (x+7)
- If you wanted to move the figure five
spaces down, you would subtract 5
from the y-coordinate of each vertex.
- Both translations can be labeled as
followed: (x, y) → (x+7, y - 5).
Preimage: Rectangle ACBD
Image: Rectangle A’C’B’D’
Translation: (x, y) → (x + 7, y - 5)

5. Reflections
- A reflection is an isometry in
which a figure and its image
have opposite orientations.
- The vertices of the preimage and
image are the same distance from
the line it is reflected across.
- Notice that when pentagon ABCDE
was reflected, the y-coordinate did
not change, and the integer value of
the x-coordinate changed.
- Example: (-6, 3) changed to (6, 3).
Preimage: Pentagon ABCDE
Image: Pentagon A’B’C’D’E’
Reflected across y-axis
A mirror is a
perfect
example of a
reflection
transformation
When looking into a mirror, the further you are, the
further away your reflection looks. The reflection
appears to be the same distance as you are from
the mirror. The object in front of the mirror is the
figure and the mirror is the line the figure is
reflecting across.

6. Reflections
- examples
- The image to the right is reflected
across the x-axis, meaning the
vertices of the preimage and image
are the same distance from the
x-axis.
- The distance from point D to
the x-axis is the same as the
distance from D’ to the x-axis
Preimage: Pentagon ABCDE
Image: Pentagon A’B’C’D’E’
Reflected across x-axis

7. Rotations
- A transformation in which
the figure circles an amount
of degrees around a point
- In order to describe a rotation, you need the
center of rotation (the point the figure is
rotating around), the angle of rotation (how
many degrees the figure is circling by).
- Rotations are either clockwise or
counterclockwise, so it needs to be stated in
the description.
- Similar to a reflection, the image is the same
distance from the center of rotation as the
preimage.
- Preimage: Quadrilateral ABCD
- Image: Quadrilateral A’B’C’
- Rotation: 90˚ counterclockwise about
the origin
*Notation: Angle of Rotation direction
about center of rotation
Real world example:
- The hands of a clock
rotate about the center
of the clock. The hands
moving from 12 to 3 is
an example of a 90˚
clockwise rotation.

8. Rotations - more
examples
- Preimage: Quadrilateral ABCD
- Image: Quadrilateral A’B’C’D’
- Rotation: 180˚ counterclockwise about
the origin
- Preimage: Quadrilateral ABCD
- Image: Quadrilateral A’B’C’D’
- Rotation: 270˚ counterclockwise about
the origin

9. Dilations
- A similarity transformation in which
the preimage and image are similar
but not congruent, which means the
transformation is not an isometry.
- Scale factor: the size change from the
preimage to the image. The scale factor is
always greater than zero.
- Center of Dilation: the point at which the
image dilates from. The distance from the
vertices of the image and the center of
dilation are proportional.
The pupil of someone’s eye
dilates based o the amount of
light the eye is exposed to. The
more light, the smaller the pupil.
The less light, the larger the
pupil. The amount of light is the
scale factor and the center of
the pupil is the center of
dilation.
- Preimage: Triangle ABC
- Image: Triangle A’B’C
- Dilation factor: 0 < x ≤ 5 (enlarging and
shrinking)
- Center of Dilation: Origin (0, 0)

10. Dilations - examples
- Preimage: Triangle ABC
- Image: Triangle A’B’C
- Dilation factor: 5
- Center of Dilation: Origin (0, 0)
- Preimage: Triangle ABC
- Image: Triangle A’B’C
- Dilation factor: x = 3
- Center of Dilation: Point C
The center of dilation can be on the figure.
This means the segments will grow outward,
with one point of the shape staying the same.
The vertices are moving along the segment’s
slope. The dilation factor is multiplied by the
length of the segment to find the new vertex.
The center of dilation can
be o the figure. For
example, when the origin is
the center and the vertices
of the figure are not on the
origin, the coordinates of
the vertices are multiplied
by the dilation factor. This
means the vertices are
moving closer/away from
the center, resulting in the
image dilating.
If the dilation factor is
greater than 1, the image
will enlarge. If the dilation
factor is less than 1, the
image will shrink. The
dilation factor can never
be zero or negative.