Construct and Apply a Sequence of Rigid Motions

New York State Common Core Math Geometry, Module 1, Lesson 19

Worksheets for Geometry

Student Outcomes

• Students begin developing the capacity to speak and write articulately using the concept of congruence. This involves being able to repeat the definition of congruence and use it in an accurate and effective way.

Construct and Apply a Sequence of Rigid Motions

Classwork

Opening

We have been using the idea of congruence already (but in a casual and unsystematic way). In Grade 8, we introduced and experimented with concepts around congruence through physical models, transparencies, or geometry software. Specifically, we had to

(1) 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; and (2) describe a sequence that exhibits the congruence between two congruent figures. (8.G.A.2)

As with so many other concepts in high school Geometry, congruence is familiar, but we now study it with greater precision and focus on the language with which we discuss it.

Let us recall some facts related to congruence that appeared previously in this unit.

1. We observed that rotations, translations, and reflections—and thus all rigid motions—preserve the lengths of segments and the measures of angles. We think of two segments (respectively, angles) as the same in an important respect if they have the same length (respectively, degree measure), and thus, sameness of these objects relating to measure is well characterized by the existence of a rigid motion mapping one thing to another. Defining congruence by means of rigid motions extends this notion of sameness to arbitrary figures, while clarifying the meaning in an articulate way.
2. We noted that a symmetry is a rigid motion that carries a figure to itself.

So how do these facts about rigid motions and symmetry relate to congruence? We define two figures in the plane as congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other.

It might seem easy to equate two figures being congruent to having same size and same shape. The phrase same size and same shape has intuitive meaning and helps to paint a mental picture, but it is not a definition. As in a court of law, to establish guilt it is not enough to point out that the defendant looks like a sneaky, unsavory type. We need to point to exact pieces of evidence concerning the specific charges. It is also not enough that the defendant did something bad. It must be a violation of a specific law. Same size and same shape is on the level of, “He looks like a sneaky, bad guy who deserves to be in jail.”

It is also not enough to say that they are alike in all respects except position in the plane. We are saying that there is some particular rigid motion that carries one to another. Almost always, when we use congruence in an explanation or proof, we need to refer to the rigid motion. To show that two figures are congruent, we only need to show that there is a transformation that maps one directly onto the other. However, once we know that there is a transformation, then we know that there are actually many such transformations, and it can be useful to consider more than one. We see this when discussing the symmetries of a figure. A symmetry is nothing other than a congruence of an object with itself.

A figure may have many different rigid motions that map it onto itself. For example, there are six different rigid motions that take one equilateral triangle with side length 1 to another such triangle. Whenever this occurs, it is because of a symmetry in the objects being compared.

Lastly, we discuss the relationship between congruence and correspondence. A correspondence between two figures is a function from the parts of one figure to the parts of the other, with no requirements concerning same measure or existence of rigid motions. If we have rigid motion 𝑇 that takes one figure to another, then we have a correspondence between the parts. For example, if the first figure contains segment 𝐴𝐵, then the second includes a corresponding segment 𝑇(𝐴)𝑇(𝐵). But we do not need to have a congruence to have a correspondence. We might list the parts of one figure and pair them with the parts of another. With two triangles, we might match vertex to vertex. Then the sides and angles in the first have corresponding parts in the second. But being able to set up a correspondence like this does not mean that there is a rigid motion that produces it. The sides of the first might be paired with sides of different length in the second. Correspondence in this sense is important in triangle similarity.

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Discussion

We now examine a figure being mapped onto another through a composition of rigid motions. To map △ 𝑃𝑄𝑅 to △ 𝑋𝑌𝑍 here, we first rotate △ 𝑃𝑄𝑅 120° (𝑅𝐷,120°) around the point, 𝐷. Then reflect the image (𝑟_𝐸𝐹 ) across 𝐸𝐹. Finally, translate the second image (𝑇𝑣 ) along the given vector to obtain △ 𝑋𝑌𝑍. Since each transformation is a rigid motion, △ 𝑃𝑄𝑅 ≅ △ 𝑋𝑌𝑍. We use function notation to describe the composition of the rotation, reflection, and translation: 𝑇𝑣 (𝐸𝐹 (𝑅𝐷, 120° (△ 𝑃𝑄𝑅))) = △ 𝑋𝑌𝑍. Notice that (as with all composite functions) the innermost function/transformation (the rotation) is performed first, and the outermost (the translation) last.

Example 1

i. Draw and label a △ 𝑃𝑄𝑅 in the space below.
ii. Use your construction tools to apply one of each of the rigid motions we have studied to it in a sequence of your choice.
iii. Use function notation to describe your chosen composition here. Label the resulting image as △ 𝑋𝑌𝑍: _______
iv. Complete the following sentences: (Some blanks are single words; others are phrases.)

△ 𝑃𝑄𝑅 is ____ to △ 𝑋𝑌𝑍 because ____ map point 𝑃 to point 𝑋, point 𝑄 to point 𝑌, and point 𝑅 to point 𝑍. Rigid motions map segments onto ____ and angles onto ____ .

Example 2

On a separate piece of paper, trace the series of figures in your composition but do NOT include the center of rotation, the line of reflection, or the vector of the applied translation. Swap papers with a partner, and determine the composition of transformations your partner used. Use function notation to show the composition of transformations that renders △ 𝑃𝑄𝑅 ≅△ 𝑋𝑌𝑍.

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