# High School: Geometry

### Congruence HSG-CO.A.3

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

By this point, students should know that transformations are movements of geometric figures on the coordinate plane. Now, it's simply a matter of building on this concept. It's just an extension of the last two standards, so hopefully it'll be kind of a no-brainer.

Students should be able to work with some geometric shapes, including parallelograms, trapezoids, and more. They should not only describe, but also identify and use points of symmetry.

In plain English, students have to illustrate how a figure is mapped onto itself by simply using a transformation without changing the shape or size of the figure. To do that, they'll need to calculate how much symmetry exists between the images, both among their lines (through reflection) and degrees (through rotational movement).

One important thing to keep in mind, though: properties such as side lengths and angle measures of the objects are key. You only have to rotate a regular hexagon 60° for it to carry onto itself, but a square needs 90°.

Both the concepts of symmetry and "carrying a geometric shape or figure onto itself" can be found in things like wallpapers, clothing, computer screensavers, and much more. If you ever need examples for symmetry, nature's got your back. And if nature's raining on your parade, then computer-generated images get pretty close to the real deal nowadays.

#### Drills

1. A regular prism has two congruent rectangles as its bases. How can one base be transformed to carry onto the other?

One base must be translated to carry onto the other base

Since the bases of the prism are congruent and oriented the same way, neither reflection nor rotation will work to carry one base onto the other. It doesn't matter how many sides a geometric shape has. We can always perform a translation and have it carry onto itself because one base just has to be "slid" across the prism. That's translation.

2. Where would you draw an imaginary line to use reflection symmetry so that a regular pentagon can carry onto itself? (By "regular pentagon," we mean one in which all angles and all side lengths are equal, not the U.S. Pentagon's less famous cousin.)

From any angle to the center of the side opposite that angle

Imagine using reflection symmetry as folding the shape along an axis. The goal is to fold along a line so that both halves carry onto each other perfectly. While (A) would work for a rectangle or square, no two lines are parallel in a pentagon. Answer (C) is incorrect because drawing line to "any other angle" would make the resulting image skewed, and (D) wouldn't work for the same reason. The only option is (B) because of angles and side lengths of a regular pentagon. If you aren't convinced, try folding one and see what works.

3. How many degrees would a regular octagon (the shape of a stop sign) need to be rotated to carry it onto itself?

45°

It takes 360° to rotate a shape back to its original starting point. Just like a square carries onto itself every 90° of rotation, regular octagon has eight equal sides, so an eighth of the total way would be enough to carry the figure onto itself. Dividing 360° by 8 gives us 45°. Now stop. Hammer time.

4. Chances are you've seen this figure, encouraging you to "Reduce, reuse, and recycle." Which of the following is true?

Rotation is the only transformation that can carry the image onto itself.

If we rotate the image one third of the way around, each arrow will land on top of the next one. Translation cannot be used, since that would mean sliding each point of the figure the same distance in the same direction, resulting in directionally opposite images. No matter how you spin it, the image also has no axis of symmetry, which means that reflection can't get the job done. Reduce, reuse, and rotate.

5. Isometry is a kind of transformation where the original image and the final image are congruent. Which of the following qualify as isometries?

Translation, rotation, and reflection

No need for your head to spin over this one. Here's the scoop: isometry simply means the points on the original image remain the same after becoming the final image. This is true regardless of the type of transformation. Since translation, rotation, and reflection all result in a final image that is congruent to the original image, they are all isometries. That means (D) is the right answer.

6. You recently realized a rectangular picture frame in your house has been upside-down this whole time. You take the photo out, fix the frame, and hang it back up. Which transformation did the picture frame undergo?

A rotation of 180°

What would you do to fix the frame? Would you flip it on its back or turn it? If you flipped it, the photo would face the wall, which wouldn't be the most constructive solution (unless all your pictures are just that hideous). If you turned it upside down completely, that would be a rotation of 180°. Whenever you're unsure, "turning" refers to rotation and "flipping" refers to reflection.

7. How many different ways can a rectangle be reflected across an axis of symmetry so that it carries onto itself?

2

Essentially, the question here is, "How many different ways can you fold a rectangle so that both halves lie on top of each other perfectly?" Just imagine folding a sheet of paper. You could fold it lengthwise and widthwise (or "hamburger" and "hot dog" style, for your inner child). If you've ever made a paper plane, you know that folding one corner to the opposite corner won't work. That means we have 2 guaranteed ways of folding a rectangle.

8. A trapezoid is a four-sided figure with two sides parallel to each other. How many axes of symmetry must a trapezoid have?

0

While trapezoids have two sides that are parallel, they don't have any other limitations as to the lengths of their sides. Some trapezoids are symmetrical and others aren't. Since the definition of a trapezoid allows for a lot of flexibility, they technically don't have to have any axis of symmetry. Some do and some don't, but none of them have to.

9. What is the order of rotational symmetry for a regular hexagon (six-sided figure)?

6

"Order of rotational symmetry" is the number of times two sides will "match" as you move around the polygon. Imagine the bottom side of a hexagon staying put as you slowly turn the shape around an entire 360°. Every time another side is carried onto the bottom one, we gain an order of rotational symmetry. There are 6 sides, which means 6 matches can be created as we make our 360° rotation. That makes an order of 6.

10. How many lines of symmetry does a circle have?

Infinite