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Common Core For Geometry

Worksheets for Geometry, Module 1, Lesson 15

Student Outcomes

- Students learn the relationship between a reflection and a rotation.
- Students examine rotational symmetry within an individual figure.

**Rotations, Reflections, & Symmetry**

Classwork

**Opening Exercise**

The original triangle, labeled A, has been reflected across the first line, resulting in the image labeled B. Reflect the image across the second line.

Carlos looked at the image of the reflection across the second line and said, “That’s not the image of triangle A after two reflections; that’s the image of triangle A after a rotation!” Do you agree? Why or why not?

**Discussion**

When you reflect a figure across a line, the original figure and its image share a line of symmetry, which we have called the line of reflection. When you reflect a figure across a line and then reflect the image across a line that intersects the first line, your final image is a rotation of the original figure. The center of rotation is the point at which the two lines of reflection intersect. The angle of rotation is determined by connecting the center of rotation to a pair of corresponding vertices on the original figure and the final image. The figure above is a 210° rotation (or 150° clockwise rotation).

**Exploratory Challenge**

**LINE OF SYMMETRY OF A FIGURE**: This is an isosceles triangle. By definition, an isosceles triangle
has at least two congruent sides. A line of symmetry of the triangle can be drawn from the
top vertex to the midpoint of the base, decomposing the original triangle into two congruent
right triangles. This line of symmetry can be thought of as a reflection across itself that takes
the isosceles triangle to itself. Every point of the triangle on one side of the line of symmetry
has a corresponding point on the triangle on the other side of the line of symmetry, given by
reflecting the point across the line. In particular, the line of symmetry is equidistant from all
corresponding pairs of points. Another way of thinking about line symmetry is that a figure
has line symmetry if there exists a line (or lines) such that the image of the figure when
reflected over the line is itself.

Does every figure have a line of symmetry?

Use your compass and straightedge to draw one line of symmetry on each figure above that has at least one line of symmetry. Then, sketch any remaining lines of symmetry that exist. What did you do to justify that the lines you constructed were, in fact, lines of symmetry? How can you be certain that you have found all lines of symmetry?

**ROTATIONAL SYMMETRY OF A FIGURE**: A nontrivial rotational symmetry of a figure is a rotation of the plane that maps the
figure back to itself such that the rotation is greater than 0° but less than 360°. Three of the four polygons above have a
nontrivial rotational symmetry. Can you identify the polygon that does not have such symmetry?

When we studied rotations two lessons ago, we located both a center of rotation and an angle of rotation. Identify the center of rotation in the equilateral triangle 𝐴𝐵𝐶 below, and label it 𝐷. Follow the directions in the paragraph below to locate the center precisely.

To identify the center of rotation in the equilateral triangle, the simplest method is finding the perpendicular bisector of at least two of the sides. The intersection of these two bisectors gives us the center of rotation. Hence, the center of rotation of an equilateral triangle is also the circumcenter of the triangle. In Lesson 5 of this module, you also located another special point of concurrency in triangles—the incenter. What do you notice about the incenter and circumcenter in the equilateral triangle?

In any regular polygon, how do you determine the angle of rotation? Use the equilateral triangle above to determine the method for calculating the angle of rotation, and try it out on the rectangle, hexagon, and parallelogram above.

**IDENTITY SYMMETRY**: A symmetry of a figure is a basic rigid motion that maps the figure back onto itself. There is a special
transformation that trivially maps any figure in the plane back to itself called the identity transformation. This
transformation, like the function 𝑓 defined on the real number line by the equation 𝑓(𝑥) = 𝑥, maps each point in the
plane back to the same point (in the same way that 𝑓 maps 3 to 3, 𝜋 to 𝜋, and so forth). It may seem strange to discuss
the do-nothing identity symmetry (the symmetry of a figure under the identity transformation), but it is actually quite
useful when listing all of the symmetries of a figure.

Let us look at an example to see why. The equilateral triangle 𝐴𝐵𝐶 on the previous page has two nontrivial rotations about its circumcenter 𝐷, a rotation by 120° and a rotation by 240°. Notice that performing two 120° rotations back-toback is the same as performing one 240° rotation. We can write these two back-to-back rotations explicitly, as follows:

- First, rotate the triangle by 120° about 𝐷: 𝑅
_{𝐷,120°}(△ 𝐴𝐵𝐶). - Next, rotate the image of the first rotation by 120°: 𝑅
_{𝐷,120°}(𝑅_{𝐷,120°}(△ 𝐴𝐵𝐶)).

Rotating △ 𝐴𝐵𝐶 by 120° twice in a row is the same as rotating △ 𝐴𝐵𝐶 once by 120° + 120° = 240°. Hence, rotating by
120° twice is equivalent to one rotation by 240°:
𝑅_{𝐷,120°}(𝑅_{𝐷,120°}(△ 𝐴𝐵𝐶)) = 𝑅_{𝐷,240°}(△ 𝐴𝐵𝐶).
In later lessons, we see that this can be written compactly as 𝑅_{𝐷,120°}
⋅ 𝑅_{𝐷,120°} = 𝑅_{𝐷,240°}
. What if we rotated by 120° one
more time? That is, what if we rotated △ 𝐴𝐵𝐶 by 120° three times in a row? That would be equivalent to rotating
△ 𝐴𝐵𝐶 once by 120° + 120° + 120° or 360°. But a rotation by 360° is equivalent to doing nothing (i.e., the identity
transformation)! If we use 𝐼 to denote the identity transformation (𝐼(𝑃) = 𝑃 for every point 𝑃 in the plane), we can
write this equivalency as follows:
𝑅_{𝐷,120°} (𝑅_{𝐷,120°}(𝑅_{𝐷,120°}(△ 𝐴𝐵𝐶))) = 𝐼(△ 𝐴𝐵𝐶).
Continuing in this way, we see that rotating △ 𝐴𝐵𝐶 by 120° four times in a row is the same as rotating once by 120°,
rotating five times in a row is the same as 𝑅_{𝐷,240°}
, and so on. In fact, for a whole number 𝑛, rotating △ 𝐴𝐵𝐶 by 120° 𝑛
times in a row is equivalent to performing one of the following three transformations:
{𝑅_{𝐷,120°}, 𝑅_{𝐷,240°}, 𝐼}.
Hence, by including identity transformation 𝐼 in our list of rotational symmetries, we can write any number of rotations
of △ 𝐴𝐵𝐶 by 120° using only three transformations. For this reason, we include the identity transformation as a type of symmetry as well.

**Exercises**

Use Figure 1 to answer the questions below.

- Draw all lines of symmetry. Locate the center of rotational symmetry.
- Describe all symmetries explicitly.

a. What kinds are there?

b. How many are rotations? (Include 360° rotational symmetry, i.e., the identity symmetry.)

c. How many are reflections? - Prove that you have found all possible symmetries.

a. How many places can vertex 𝐴 be moved to by some symmetry of the square that you have identified? (Note that the vertex to which you move 𝐴 by some specific symmetry is known as the image of 𝐴 under that symmetry. Did you remember the identity symmetry?)

b. For a given symmetry, if you know the image of 𝐴, how many possibilities exist for the image of 𝐵?

c. Verify that there is symmetry for all possible images of 𝐴 and 𝐵.

d. Using part (b), count the number of possible images of 𝐴 and 𝐵. This is the total number of symmetries of the square. Does your answer match up with the sum of the numbers from Exercise 2 parts (b) and (c)?

**Relevant Vocabulary**

**REGULAR POLYGON**: A polygon is regular if all sides have equal length and all interior angles have equal measure.

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