# High School: Geometry

### Congruence HSG-CO.A.4

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Students should be able to recognize and visualize transformations of geometric shapes and not only reflect, rotate, and translate shapes, but also combine these three different transformations. By defining and describing these transformations in reference to angles, circles, lines and line segments, students should gain a deeper understanding of these transformations and when these operations apply to real world situations.

Show how rotations, reflections, and translations are drawn, but also to use polygons in order to perform a few of these transformations. Then, explain how each step meshes with the definition of that transformation. The more examples, the merrier!

Also, remind students that when a geometric shape undergoes a rigid transformation, its angles cannot change. Rigid transformations do not affect length, area, or angle measure of a geometric shape. They do, however, affect your flexibility in yoga class.

Students should use the three transformations listed to explore and prove geometric properties. The best way to analyze these transformations in terms of angles, circles, line segments, and perpendicular and parallel lines is to consider each individually.

For example, your students are driving (and definitely not texting) and approaching a four-way intersection. The two streets will be perpendicular to one another. If we picture a line down the middle of one of the streets, either horizontally or vertically, and we imagine folding the streets over each other, we will have a reflection transformation.

The angles and the angles are all mirrored, but due to the 90° angles all around, they are congruent either way. Of course, actually folding streets over would be some crazy Inception-style infrastructure, but you get the point.

#### Drills

1. Look at the angle formed by each spoke stretching to the outer edge of this wheel. What is the figure's order of rotational symmetry?

8

The wheel can be turned 8 times to create the maximum number of matches among the identical angles around the outer edge. An order of 4 would mean turning the wheel and missing every other match, so (A) is incorrect; and (C) is incorrect for the opposite reason: half the "matches" wouldn't be matches at all. Although the outer edge of the wheel is a circle, the spokes limit the order of rotational symmetry to 8 rather than infinity.

2. Below are four road signs. Which has the greatest order of rotational symmetry?

This is partly defined by its line segments, its angles, and the circular arcs on the face of the sign. Just imagine rotating each of the signs a full 360° and imagine how many times the image repeats itself. The three arrows in (A) indicate an order of 3, while (B) has an order of 2. Although (C) is symmetrical, it has no rotational symmetry, which gives it an order of 1. While the shape of the stop sign has an order of 8, the word "STOP" gives the sign an order of 1 as well. Our highest is (A).

3. Some parking lots have straight parking spaces and others have diagonal ones. If the two types of parking spaces were to be reflected, only the straight parking space's reflection could be carried onto its original image. Why?

The straight parking space has perpendicular lines

If you imagine "folding" a straight parking space, you will end up with a perfect reflection transformation. The reason this is possible is because all lines on the right line up precisely with those on the left because of the right angles at the 2 lower corners of each. When reflected, the obtuse and acute angles of the diagonal parking space are switched, so they don't line up with one another. The lengths of the lines in the parking spaces have nothing to do with the transformation. It's all about the right angles.

4. What is the minimum number of degrees that a pentagram can be rotated so that it matches with itself?

72°

Since a pentagram has 5 equal parts, its order of rotational symmetry is 5. That means that out of 360° it will repeat itself 5 times. In other words, every 3605 = 72°, the pentagram will have rotated enough to look like its original image. Remember to look at the order of rotational symmetry from the proper angle, however many degrees that is.

5. What is the order of rotational symmetry for a regular polygon with n sides?

n

A square has 4 sides and an order of 4. A regular octagon has 8 sides and an order of 8. A regular polygon with 27 sides will have an order of 27. We're getting the hunch that the number of sides of a regular polygon has to do with the order of rotational symmetry. Well, maybe more than a hunch.

6. Will the rotation of a pair of parallel lines always result in another pair of parallel lines?

Yes

The key here is that we're rotating the pair of lines as though it were one image. If the lines are parallel in the original image, any rigid transformation will keep the lines parallel to each other. Rotating the parallel lines at 180° and 360° would keep the lines in the new image parallel to those in the original one, but this isn't necessary to keep the lines in the image parallel to each other. They'll always be that way because rotation is a rigid transformation.

7. A square is translated five units to the right. What will be the angles of the new square?

90°

Remember that translation, just like the other rigid transformations, conserves angle measures. Translating a figure moves the entire thing over, keeping all the angles and side lengths the same. The square started off with 4 angles, each at 90° and it should stay that way.

8. Which of the following do rigid transformations not do?

Conserve the ability to carry the new figure into the original figure

You've probably gathered by now that reflections often do not allow for a new figure to be carried onto the original figure. Just imagine your hands as mirror images of each other, and try to carry one into the other. Can't do it without flipping 'em, can you? While all the lengths of your fingers and the angles between them are the same, they can't be carried onto one another. Not without even more transformations, anyway.

9. Which of the following is true about circles and rigid transformations?

A circle can be carried onto itself regardless of any rigid transformation performed on it.

We should immediately know that (A) and (D) aren't true because rigid transformations conserve angles and lengths. It's practically in the name. While (C) might seem tempting, circles are actually identical with even the smallest degree of rotation—even after 0.0001°. What is true is (B) because circles have infinite symmetry.

10. What is the difference between changing an angle and changing an angle of rotation?

An angle is changed if the two rays that form the angle are moved, while an angle of rotation is changed if an image is moved about a central point.