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9-12 Geometry Unit 1 Lesson 17

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K-5 Math 6-8 Math •9-12 Math

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Algebra 1 •Geometry Algebra 2 Extra Support Materials for Algebra 1

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Lesson 17

17.1 Warm-up 17.2 Activity 17.3 Activity Student Lesson Summary

Unit 1, Lesson 17

Working with Rigid Transformations

Geometry

17.1

Warm-up

Math Talk: From Here to There

Segment is the perpendicular bisector of segment . Find each transformation mentally.

A transformation that takes to .
A transformation that takes to .
A transformation that takes to .
A transformation that takes to .

<p>Segments AB and CD intersect at point M. CD is the perpendicular bisector of AB.</p>

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17.2

Activity

Card Sort: How Did This Get There?

Your teacher will give you a set of cards that show transformations of figures. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
For each card with a rigid transformation, write a sequence of rotations, translations, and reflections to get from the original figure to the image. Be precise.

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17.3

Activity

Reflecting on Reflection

Diego says, “I see why a reflection could take to , but I’m not sure where the line of reflection is. I’ll just guess.”

How could Diego see that a reflection could work without knowing where the line of reflection is?
How could Diego find an exact line of reflection that would work?

Student Lesson Summary

If two figures are congruent, we can always find a rigid transformation that takes one onto the other.

Look at congruent figures and . It looks like might be a translation, rotation, and reflection of . But is there a way to describe a sequence of transformations without guessing where the line of reflection might be?

<p>Two congruent triangles labeled ABC and DEF.</p>

Our goal is to take onto . Then we want to take the image of onto without moving and . Finally, we need to take the image of onto without moving any of the matching points.

We can start with translation: Translate triangle by the directed line segment from to .

<p>Three congruent triangles, DEF, D’E’F’ and ABC.</p>

<p> </p>

Now, a pair of corresponding points coincides. Is there a transformation we could use to take onto that leaves and in place? Rotations have a fixed point, so rotate triangle by angle using point as the center.

<p>Four congruent triangles, DEF, D’E’F’, D”E”F” and ABC.</p>

Now, two pairs of corresponding points coincide. Reflecting across line will take onto , which is what we were trying to do. We know and won’t move since points on the line of reflection don't move. How do we know will end up on ? Since the triangles are congruent, and are the same distance from the line of reflection.

It is always possible to describe transformations using existing points, angles, and segments. It could take an extra step, but we can be confident that transformations work if we don't guess where the line of reflection or center of rotation might be.

Glossary

None

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