# High School: Geometry

### Congruence HSG-CO.A.2

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

In today's world, it seems more and more people are trying to transform themselves in some way. Whether they want to transform their inner selves through self-help books and meditation or their outer selves with exercise and more cosmetic surgery than Joan Collins can handle, we shouldn't forget that shapes sometimes need a change as well. These changes are called transformations.

No, not Transformers. They're cool and all, but they're not the same.

A transformation means somehow altering a shape on the coordinate plane. Whether we move, flip, stretch, shrink, or turn the shape, we're performing a transformation. Unfortunately, an Extreme Makeover doesn't count when it comes to shapes.

The three main types of transformations—translations, reflections, and rotations—are called "rigid" because they preserve the distance and angles of their shapes.

Though it means something much different in your students' foreign language classes, translation means moving a shape in one direction. They can think of it as "sliding" that shape. No turning, no flipping, no complex movements. Just a straight slide. The shape stays exactly the same, but in a new place.

Just as it sounds, rotation is a circular movement. The shape moves around a central point. It's like turning a dial in one direction or the other, causing the object to stay where it is, but spin clockwise or counterclockwise. After 360°, though, you end up right where you started.

Students should think of reflection as looking at a shape in a mirror. Everything about the shape is the same, except one is the mirror image of the other. If imagining a mirror is strange or difficult, your right and left hand are reflections of each other, too.

Students should know that when any of these transformations are done in the coordinate plane, they can be described as functions that take points in the plane as inputs and give other points as outputs. The best way to see this is through coordinates of a simple shape, like a triangle. Moving up one unit means we add 1 to every y coordinate. Our input is y and our output is y + 1. Easy peasy.

We've been talking about transformations that move shapes without altering the shape itself—only the coordinates on which it sits. In other words, these transformations "preserve distance and angle," but there are some transformations that change them. This happens when multiplication is used, as opposed to addition or subtraction.

When you multiply, the distances between lines of the shape on the coordinate plane are lengthened. Multiply the x coordinates by 2, and the shape grows twice as wide; multiply the y coordinates by 2, and the shape grows twice as tall. This is called a "stretch" and tell your students to be careful with them. Otherwise, we may end up with gigantic shapes that'll want to take over the world.

On the other hand, if you multiply the coordinates by a number less than 1, the transformation is called compression. If compression were made into a Hollywood movie, it'd be called, "Honey, I Shrunk the Shapes."

Students should know how to move a given figure around the coordinate plane according to the different types of transformations. They should also represent a translation as a function using coordinates and understand the difference between transformations that preserve distance and angle and those that do not.

In no time, they'll be transforming themselves into the best geometry students you could hope for! Awww.

#### Drills

1. If you ride the elevator from the lobby of the Empire State Building to the very top, is this motion a transformation, a translation, or a rotation?

Transformation and translation

First of all, both translations and rotations are types of transformations. So that rules out (A) and (B). When you take the elevator, your orientation stays the same; you don't turn or spin, except in the unlikely event that the elevator turns upside-down. You slide upwards until you reach the top floor. So, while you wouldn't normally use the term in conversation, it is correct to say the elevator has "translated."

2. We already know that geometric objects on a coordinate plane are meant to represent and connect to points in space. So, using outer space as our setting, what is an example of a rotation?

The earth turning on its axis

Every 24 hours, the earth makes one complete turn on its own axis. It moves around the same center point and does not change shape or size. That's the definition of rotation. In (A), there is no mention of Saturn turning or rotating in any way. Even if it did, the rings are rotating, not Saturn itself. The asteroid in (C) is moves in a straight line, which is translation, not rotation. It's also likely changing shape as it flies through space, too. A lunar eclipse, in which our moon passes into the earth's shadow, simply forms a line segment between the two. The only rotation, then, is (B).

3. What is the formal term for the line that an object is reflected across?

Axis of symmetry

Though (A) is logical, it is not a term used in transformations or congruence. Two reflected images are often called, "laterally inverted," but this term is not used to describe the line between the two. A median is the segment connecting the vertex of one angle in a triangle to the midpoint of its opposite side. It doesn't have to do with reflection necessarily. The right answer is (C).

4. What kind of transformation turns ΔABC into ΔDEF?

Reflection

It should be clear from the image that a translation would not be enough to carry ΔABC into &DEF. A rotation would get us closer, but the points B and E wouldn't match up, exactly. The only option left is reflection. If you're uncertain about why this is true, try translating and rotating your right hand onto your left. The only way to get them to match up is to "flip" or reflect it.

5. A community wants to move a skateboard park for safety and noise reasons. The volunteers decide to move the skateboard park 128 feet east and 52 feet south. Assuming the positive y-axis on a coordinate plane as north, which function represents the translation coordinates of the skateboard park?

(x, y) → (x + 128, y – 52)

The question tells us that the skateboard park is moved 128 feet along the east-west axis (that is, along the x-axis) and the 52 feet along the north-south axis (that is, the y-axis). That eliminates (A) right off the bat. If north is in the positive y direction, east should be in the positive x direction. A translation of 128 feet to the east means x + 128, and a translation of 52 feet south means y – 52. Therefore, our answer is (B).

6. The pool of a health club undergoing renovation is being moved from the center of the bottom floor to the far right roof deck. If they want to move the pool up 8 stories and to the right 6 yards, which of the following represents the job the construction workers need to do?

6 units in the +x direction and 8 units in the +y direction

The construction workers must move up 8 floors, meaning +8 along the y-axis of our coordinate plane. They must also move to the right 6 yards along our x-axis (in the positive direction). That not only means that (A) is correct, but also that any option with a negative sign can be instantly eliminated.

7. Is it possible for translation, rotation, and reflection to produce the same image?

Yes, this is possible if the original image is in some way symmetrical

If (A) were the right answer, we wouldn't need three different types of transformations! They'd all be the same. If we imagine a circle as the image, a shifted, turned, and reflected circle would still be the exact same circle. However, if we shift, reflect, and turn a square, it will be identical to the original square as well (provided the turns are 90°, 180°, 270°, or 360° turns). Since it's true for shapes other than circles, (B) is the right answer.

8. Resizing occurs when a shape is compressed or dilated in size, but still retains its unique shape. Which of the following is true about resizing?

Resizing creates an image that is similar or proportional to the one before it.

Since resizing includes dilation (growing) and compression (shrinking). That means we can create an image smaller than the one before it. Transformation can happen anywhere in the coordinate plane, so the resized image doesn't have to be placed on the other side of the y-axis. The definition of congruence tells us that an image larger or smaller than another isn't going to be identical to it, so (C) is our only option.

9. Which of the following best describes a stretch? (No, we don't mean a yoga stretch.)

A transformation in which all distances on the coordinate plane are lengthened by multiplying either all x coordinates and/or all y coordinates by a factor greater than 1

A stretch is a form of dilation in which all distances on the plane are lengthened by a single factor through multiplication by a single factor greater than 1. It can be done to either the x coordinates or the y coordinates or both. Answer (A) is incorrect because it restricts the operation to only the x coordinates. Answer (B) is incorrect as it refers to the shape being multiplied by a factor less than 1, which would compress the image rather than stretch it. Stretches don't conserve distance, but reflections are rigid transformations that do.

10. Which of the following is the best way to describe a geometric translation?