# Lesson 17

Working with Rigid Transformations

### Lesson Narrative

In previous grades, students describe a sequence of rigid transformations that exhibits the congruence between two figures. To prepare students for future congruence proofs, this lesson asks students to come up with a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. As the focus shifts to sequences of transformations between figures with more general characteristics rather than specific measurements, encourage students to explain how they know that their sequences will cause certain points or lines to coincide. When students consider how generalizable a strategy for defining sequences of rigid transformation is, they are looking for the structures of pairs of congruent figures (MP7).

### Learning Goals

Teacher Facing

• Compare and contrast (orally) diagrams of transformations.
• Comprehend that the notation $A'$ represents the image of point $A$.
• Explain (orally and in writing) a sequence of transformations that take given points to another set of points.

### Student Facing

• Let’s compare transformed figures.

### Student Facing

• I can describe a transformation that takes given points to another set of points.

Building On

Building Towards

### Glossary Entries

• assertion

A statement that you think is true but have not yet proved.

• congruent

One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

• directed line segment

A line segment with an arrow at one end specifying a direction.

• image

If a transformation takes $$A$$ to $$A'$$, then $$A$$ is the original and $$A'$$ is the image.

• line of symmetry

A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

• reflection

A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line.

In the figure, $$A'$$ is the image of $$A$$ under the reflection across the line $$m$$.

• reflection symmetry

A figure has reflection symmetry if there is a reflection that takes the figure to itself.

• rigid transformation

A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

• rotation

A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

$$P'$$ is the image of $$P$$ after a counterclockwise rotation of  $$t^\circ$$ using the point $$O$$ as the center.

​​​​​Quadrilateral $$ABCD$$ is rotated 120 degrees counterclockwise using the point $$D$$ as the center.

• rotation symmetry

A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as $$0^\circ$$ and $$360^\circ$$ that leave every point on the figure where it is.)

• symmetry

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

• theorem

A statement that has been proved mathematically.

• translation

A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

In the figure, $$A'$$ is the image of $$A$$ under the translation given by the directed line segment $$t$$.