Main content

## 8th grade (Eureka Math/EngageNY)

### Unit 2: Lesson 1

Topic A: Definitions and properties of the basic rigid motions- Introduction to geometric transformations
- Identifying transformations
- Identify transformations
- Translations intro
- Translating points
- Translate points
- Translating shapes
- Translating shapes
- Translate shapes
- Determining translations
- Determine translations
- Rotations intro
- Rotating points
- Rotate points (basic)
- Determining rotations
- Determine rotations (basic)
- Reflecting points
- Reflect points
- Reflecting shapes
- Reflecting shapes
- Reflect shapes
- Determining reflections
- Determine reflections
- Translations review
- Rotations review
- Reflections review

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# Reflecting shapes

Learn how to find the image of a given reflection.

In this article we will find the images of different shapes under different reflections.

## The line of reflection

A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.

The line of reflection can be defined by an equation or by two points it passes through.

## Part 1: Reflecting points

### Let's study an example of reflecting over a horizontal line

We are asked to find the image A, prime of A, left parenthesis, minus, 6, comma, 7, right parenthesis under a reflection over y, equals, 4.

#### Solution

**Step 1:**Extend a perpendicular line segment from A to the reflection line and measure it.

Since the reflection line is perfectly horizontal, a line perpendicular to it would be perfectly vertical.

**Step 2:**Extend the line segment in the same direction and by the same measure.

**Answer:**A, prime is at left parenthesis, minus, 6, comma, 1, right parenthesis.

### Your turn!

#### Practice problem

#### Challenge problem

### Let's study an example of reflecting over a diagonal line

We are asked to find the image C, prime of C, left parenthesis, minus, 2, comma, 9, right parenthesis under a reflection over y, equals, 1, minus, x.

#### Solution

**Step 1:**Extend a perpendicular line segment from C to the reflection line and measure it.

Since the reflection line passes exactly through the diagonals of the unit squares, a line perpendicular to it should pass through the other diagonal of the unit square. In other words,

*lines with slopes start text, 1, end text and start text, negative, 1, end text are always perpendicular.*For convenience, let's measure the distance in "diagonals":

**Step 2:**Extend the line segment in the same direction and by the same measure.

**Answer:**C, prime is at left parenthesis, minus, 8, comma, 3, right parenthesis.

### Your turn!

#### Practice problem

#### Challenge problem

## Part 2: Reflecting polygons

### Let's study an example problem

Consider rectangle E, F, G, H drawn below. Let's draw its image E, prime, F, prime, G, prime, H, prime under a reflection over the line y, equals, x, minus, 5.

#### Solution

When we reflect a polygon, all we need is to perform the reflection on all of the vertices (this is similar to how we translate or rotate polygons).

Here are the original vertices and their images. Notice that E, F, and H were on an opposite side of the reflection line as G. The same is true about their images, but now they switched sides!

Now we simply connect the vertices.

### Your turn!

#### Problem 1

#### Problem 2

## Want to join the conversation?

- I could really use Sal making a video about this, what’s written on this doc is really confusing.

Sometimes they explain things that are pretty basic and other times more complicated things they’ll just assume that we know them even though we haven’t covered it/them yet.

For instance I don’t understand what they mean when referring to the reflection line

Y=1-x

Y=x+2

Y= x-5(17 votes)- The reflection line is the line that you are reflecting over. Y=mx+b is just the basic slope-intercept equation. If you don't understand slope -intercept, I recommend watching the videos Khan provides in the algebra courses. Since geometry tends to be taught after algebra in some cases, I think it's why they didn't explain it more in depth. Hope this helps!(4 votes)

- why cant there be a video on this i dont understand it but a video would help(8 votes)
- There is a part that says "I want to see Sal doing a similar question" which helped me since I was having trouble.(2 votes)

- isn't there an algebraic formula for this ?(3 votes)
- When you reflect a point across the line y = x, the x-coordinate and y-coordinate change places. If you reflect over the line y = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed). the line y = x is the point (y, x). the line y = -x is the point (-y, -x).(8 votes)

- how do you know what way to reflect it(4 votes)
- Seriously, this math stumps me(4 votes)
- can you please eplain reflection to me please again(3 votes)
- Reflection is taking an object and flipping it around over a line (which you could think of as a mirror).(3 votes)

- This is a probably a stupid question, but i totally do not get why -- for example -- in this problem:

Draw the image of triangle MNO under a reflection over: y = -1 -x

I don't get what y = -1 -x means, is it a coordinate or is it comparing y to x, i just don't understand, same goes for the other problems:

What is the image of (-12, 12) under a reflection over line y = x. I can usually solve the problem, but i feel like i still need to understand what is means.(3 votes)- No questions are stupid! y=-1-x and y=x are both lines. When you reflect a point, it is an equal distance away from the line as your original point. For instance, (-12,12) reflected over y=x would be (12,-12). I hope this clears things up!(2 votes)

- What do you do when the line doesn't meet consistent points? Like in the line y-2x-5(3 votes)
- what in the world if they say this is geometry then how do u do it I am usually good at this kind of thing it is passed through the family(3 votes)
- That was a little confusing? try this https://www.khanacademy.org/math/basic-geo/basic-geo-transformations-congruence/basic-geo-reflections/e/performing-reflections-on-the-coordinate-plane or https://www.khanacademy.org/math/basic-geo/basic-geo-transformations-congruence/basic-geo-reflections/v/reflecting-shapes(3 votes)