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 .
17.2
Activity
Card Sort: How Did This Get There?
Standards Alignment
Building On
8.G.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
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.
17.3
Activity
Reflecting on Reflection
Standards Alignment
Building On
G-CO.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
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?
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 .
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.
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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Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
