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Sequences of Rigid Motions




 

Videos and solutions to help Grade 8 students describe a sequence of rigid motions to map one figure onto another.

New York State Common Core Math Grade 8, Module 2, Lesson 10.

Download New York State Common Core Math Grade 8, Module 2, Lesson 10 Worksheets (PDF)

Related Topics:
Lesson Plans and Worksheets for Grade 8
Lesson Plans and Worksheets for all Grades
More Lessons for Grade 8
Common Core For Grade 8

Classwork

Example 1
So far we have seen how to sequence translations, sequence reflections, and sequence translations and reflections. Now that we know about rotation, we can move geometric figures around the plane by sequencing a combination of translations, reflections and rotations.

Let's examine the following sequence:
• Let E denote the ellipse in the coordinate plane as shown.

• Let Translation1 be the translation along the vector v from (1, 0) to (-1, 1), let Rotation2 be the 90 degree rotation around (-1, 1), and let Reflection3 be the reflection across line L joining (-3, 0) and (0, 3). What is the Translation1(E) followed Rotation2(E) by the followed by the Reflection3(E)?

• Which transformation do we perform first, the translation, the reflection or the rotation? How do you know? Does it make a difference?

• Which transformation do we perform next?

• Now the only transformation left is Reflection3. So now we let E3 be the image of E after the Translation1(E) followed by the Rotation2(E) followed by the Rotation3(E)

Exercises

1 - 3. In the following picture, triangle ABC can be traced onto a transparency and mapped onto triangle A'B'C'. Which basic rigid motion, or sequence of, would map one triangle onto the other?

4. In the following picture, we have two pairs of triangles. In each pair, triangle ABC can be traced onto a transparency and mapped onto triangle A'B'C'.

Which basic rigid motion, or sequence of, would map one triangle onto the other?

5. Let two figures ABC and A'B'C' be given so that the length of curved segment AC = the length of curved segment A'C', |∠ B| = |∠ B'| = 80 ° and |AB| = |A'B'| = 5. With clarity and precision, describe a sequence of rigid motions that would map figure ABC onto figure A'B'C'.

Composition of Rigid Motions
This video shows how we can move geometric figures around the plane by sequencing a combination of translations, reflections and rotations.
We start with two identical figures. What sequence of basic rigid motions maps the left H exactly onto the right H so that all corresponding angles and segments coincide?



How to precisely describe a set of rigid motions to map one figure onto another?
We know that the order in which transformations are performed makes a difference. For each of the following exercises, we will go into more detail on how exactly to describe rigid motions that will map one figure onto another.


 

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