6th Grade Math Textbook, Fundamentals

378 Chapter 13

Enrichment:

Combining Transformations

Objective To apply two or more transformations to a figure on the coordinate plane

You have learned about several kinds of
transformations: translations, reflections,
rotations, and dilations. The first three are
sometimes called congruence transformations,
because the image is congruent to the original
figure (the pre-image). Note that a dilation
does not result in a congruent image, but
rather in an image that is similar to the
pre-image.

In this lesson, you will explore combinations
of transformations. That is, you will investigate
what happens when you apply a transformation
to a figure and then apply another transformation
to the image.

What happens when equilateral BOPat the
right is reflected first over the x-axis and then
over the y-axis?

Draw your own triangle on a coordinate plane and
perform these two reflections on it. Label the first
image BOPand the final image BOP.

You can see that BOPis congruent to BOPand to BOP.
The only change has been in the positions of the triangles.

There are other combinations of transformations that will transform

the pre-image BOPinto BOP. Which of the following

combinations results in the image shown in Quadrant III?

• Reflect BOPfirst over the y-axis and
  then again over the x-axis.


• Reflect BOPover the x-axis and then
  translate the image 6 units to the left.


• Translate BOPsix units to the left and
  then reflect the image over the x-axis.


• Rotate BOP180° about the origin.

An easy way to test all these combinations of transformations

is to cut out a tracing of the pre-image triangle and move it

around on the grid above. Include the labels for the vertices

in your tracing. You will find that the second and third

combinations have the letters Oand Preversed, so they

do not produce a match.

(You might try testing these same transformation combinations

with a triangle that is not isosceles. Which ones still “work”?)

B

P O

B‘

O‘

B“

P“

y

x

Quadrant II Quadrant I

Quadrant III Quadrant IV

O“ P‘