geometry reflections homework

Reflections are everywhere ... in mirrors, glass, and here in a lake. ... what do you notice ?

Every point is the same distance from the central line !

... and ...

The reflection has the same size as the original image.

The central line is called the Mirror Line :

Can A Mirror Line Be Vertical?

Yes. Here my dog "Flame" shows a Vertical Mirror Line (with a bit of photo editing).

In fact Mirror Lines can be in any direction . Imagine turning the top image in different directions:

A reflection is a flip over a line

You can try reflecting some shapes about different mirror lines here:

"how do i do it myself".

Just approach it step-by-step. For each corner of the shape:

It is common to label each corner with letters, and to use a little dash (called a Prime ) to mark each corner of the reflected image.

Here the original is ABC and the reflected image is A'B'C'

Some Tricks

When the mirror line is the y-axis we change each (x,y) into (−x,y)

Fold the Paper

And when all else fails, just fold the sheet of paper along the mirror line and then hold it up to the light !

Calcworkshop

Reflection Rules How-To w/ 25 Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

What are the Reflection Rules?

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

That’s what today’s geometry lesson is all about.

You’re going to learn how to find the line of reflection, graph a reflection in a coordinate plane, and so much more.

So let’s get started!

A transformation that uses a line that acts as a mirror, with an original figure ( preimage ) reflected in the line to create a new figure ( image ) is called a reflection.

A reflection is sometimes called a flip or fold because the figure is flipped or folded over the line of reflection to create a new figure that is exactly the same size and shape.

Reflection Example

What is important to note is that the line of reflection is the perpendicular bisector between the preimage and the image. Thus ensuring that a reflection is an isometry, as Math Bits Notebook rightly states.

Reflection on a Coordinate Plane

Reflection over x axis.

Reflection Across the X-Axis

Reflection Over Y Axis

Reflection Across the Y-Axis

Reflection Across Y=X

Reflection Over Y=X

Reflection Across Y=-X

Reflection Over Y = -X

In order to define or describe a reflection, you need the equation of the line of reflection. The four most common reflections are defined below:

Common Reflections About the Origin

Reflection Symmetry

Additionally, symmetry is another form of a reflective transformation. When a figure can be mapped (folded or flipped) onto itself by a reflection, then the figure has a line of symmetry.

For example, the image of a heart has one line of symmetry, as we can fold the heart in half to create the same shape. Similarly, the example of the equilateral triangle below has three lines of symmetry, as we can fold the triangle along these lines to create equal halves.

1 Line of Symmetry

3 Lines of Symmetry

Glide Reflection

A glide reflection is a composite transformation where we translate (glide) and then reflect a figure in successive steps. But what is super cool about glide reflections is that as long as the translation is parallel to the line of reflection, it doesn’t matter which transformation you perform first. So that means we can slide then flip, or we can flip then slide.

Glide Reflection Examples

And did you know that reflections are used to help us find minimum distances?

Now we all know that the shortest distance between any two points is a straight line, but what would happen if you need to go to two different places?

For example, imagine you and your friend are traveling together in a car. You need to go to the grocery store and your friend needs to go to the flower shop. Where should you park the car minimize the distance you both will have to walk?

The answer is found using reflections!

Next, you’ll learn how to:

• Draw reflections.
• Describe the reflection by finding the line of reflection.
• Determine the number of lines of symmetry.
• Find a point on the line of reflection that creates a minimum distance.

Video – Lesson & Examples

• Introduction to Reflections
• 00:00:43 – Properties of Reflections: Graph and Describe the Reflection (Examples #1-4)
• Exclusive Content for Member’s Only
• 00:10:53 – How to find the line of reflection (Examples #5-7)
• 00:17:45 – Graph the given reflection in the coordinate plane (Examples #8-13)
• 00:25:02 – Determine the number of lines of symmetry (Examples #14-17)
• 00:30:22 – Determine how a square piece of paper will look once unfolded (Examples #18-20)
• 00:35:42 – Glide Reflections and the Composition Theorem (Examples #21-22)
• 00:44:53 – Overview of how we can Optimize with Geometry
• 00:52:16 – Finding the minimum distance using reflections (Examples #23-25)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Reflections

Related Topics: Lesson Plans and Worksheets for Geometry Lesson Plans and Worksheets for all Grades More Lessons for Geometry Common Core For Geometry

New York State Common Core Math Geometry, Module 1, Lesson 14

Worksheets for Geometry

Student Outcomes

• Students learn the precise definition of a reflection.
• Students construct the line of reflection of a figure and its reflected image.
• Students construct the image of a figure when provided the line of reflection.

Exploratory Challenge

Think back to Lesson 12 where you were asked to describe to your partner how to reflect a figure across a line. The greatest challenge in providing the description was using the precise vocabulary necessary for accurate results. Let’s explore the language that yields the results we are looking for.

△ 𝐴𝐵𝐶 is reflected across 𝐷𝐸 and maps onto △ 𝐴′𝐵′𝐶′.

Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′. What do you notice about these perpendicular bisectors?

Label the point at which 𝐴𝐴′ intersects 𝐷𝐸 as point 𝑂. What is true about 𝐴𝑂 and 𝐴′𝑂? How do you know this is true?

You just demonstrated that the line of reflection between a figure and its reflected image is also the perpendicular bisector of the segments connecting corresponding points on the figures. In the Exploratory Challenge, you were given the pre-image, the image, and the line of reflection. For your next challenge, try finding the line of reflection provided a pre-image and image.

Construct the segment that represents the line of reflection for quadrilateral 𝐴𝐵𝐶𝐷 and its image 𝐴′𝐵′𝐶′𝐷′.

What is true about each point on 𝐴𝐵𝐶𝐷 and its corresponding point on 𝐴′𝐵′𝐶′𝐷′ with respect to the line of reflection?

Notice one very important fact about reflections. Every point in the original figure is carried to a corresponding point on the image by the same rule—a reflection across a specific line. This brings us to a critical definition:

REFLECTION : For a line 𝑙 in the plane, a reflection across 𝑙 is the transformation 𝑟𝑙of the plane defined as follows:

• For any point 𝑃 on the line 𝑙, 𝑟𝑙 (𝑃) = 𝑃, and
• For any point 𝑃 not on 𝑙, 𝑟𝑙 (𝑃) is the point 𝑄 so that 𝑙 is the perpendicular bisector of the segment 𝑃𝑄.

If the line is specified using two points, as in 𝐴𝐵 , then the reflection is often denoted by 𝑟𝐴𝐵. Just as we did in the last lesson, let’s examine this definition more closely:

• A transformation of the plane—the entire plane is transformed; what was once on one side of the line of reflection is now on the opposite side;
• 𝑟𝑙(𝑃) = 𝑃 means that the points on line 𝑙 are left fixed—the only part of the entire plane that is left fixed is the line of reflection itself;
• 𝑟𝑙(𝑃) is the point 𝑄—the transformation 𝑟𝑙 maps the point 𝑃 to the point 𝑄;
• The line of reflection 𝑙 is the perpendicular bisector of the segment 𝑃𝑄—to find 𝑄, first construct the perpendicular line 𝑚 to the line 𝑙 that passes through the point 𝑃. Label the intersection of 𝑙 and 𝑚 as 𝑁. Then locate the point 𝑄 on 𝑚 on the other side of 𝑙 such that 𝑃𝑁 = 𝑁𝑄.

Examples 2–3 Construct the line of reflection across which each image below was reflected

• You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore methods for constructing the reflected image itself. The first few steps are provided for you in this next stage.

Example 4 The task at hand is to construct the reflection of △ 𝐴𝐵𝐶 over ̅𝐷𝐸̅̅̅. Follow the steps below to get started; then complete the construction on your own.

• Construct circle 𝐴: center𝐴, with radius such that the circle crosses ̅𝐷𝐸̅̅̅ at two points (labeled 𝐹 and 𝐺).
• Construct circle 𝐹: center 𝐹, radius 𝐹𝐴 and circle 𝐺: center 𝐺, radius 𝐺𝐴. Label the (unlabeled) point of intersection between circles 𝐹 and 𝐺 as point 𝐴′. This is the reflection of vertex 𝐴 across ̅𝐷𝐸̅̅̅.
• Repeat steps 1 and 2 for vertices 𝐵 and 𝐶 to locate 𝐵′ and 𝐶′.
• Connect 𝐴′, 𝐵′, and 𝐶′ to construct the reflected triangle. Things to consider: When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection itself? Why did the construction above work?

Example 5 Now try a slightly more complex figure. Reflect 𝐴𝐵𝐶𝐷 across 𝐸𝐹̅̅̅̅.

Lesson Summary

• A reflection carries segments onto segments of equal length.
• A reflection carries angles onto angles of equal measure.

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Transformation Worksheets: Translation, Reflection and Rotation

Exercise this myriad collection of printable transformation worksheets to explore how a point or a two-dimensional figure changes when it is moved along a distance, turned around a point, or mirrored across a line. Encompassing basic transformation practice on slides, flips, and turns, and advanced topics like translation, rotation, reflection, and dilation of figures on coordinate grids, these pdf worksheets on transformation of shapes help students of grade 1 through high school sail smoothly through the concept of rigid motion and resizing. Access some of these worksheets for free!

Printing Help - Please do not print transformation worksheets directly from the browser. Kindly download them and print.

» Slide, Flip and Turn

» Rotation

» Translation

» Reflection

» Dilation with Center at Origin

» Dilation with Center not at Origin

Identify the Transformation

In these worksheets identify the image which best describes the transformation (translation, reflection or rotation) of the given figure. Ideal for grade 5 and grade 6 children.

• Download the set

Write the Type of Transformation

Each grid has the figure and the image obtained after transformation. Write, in each case the type of transformation undergone. Recommended for 6th grade and 7th grade students.

Transformation of Points: Multiple Choices

Rotate, reflect and translate each point following the given rules. Grade 7 students should choose the correct image of the transformed point.

Multiple Choices: Transformation

The coordinates of a point are given. Perform the required transformation and check mark the correct choice.

Transformation of Shapes

Translate, reflect or rotate the shapes and draw the transformed image on the grid. Each printable worksheet has eight practice problems.

Transformation of Triangles

Draw the transformed image of each triangle. The type of transformation to be performed is described above each question.

Transformation of Quadrilaterals

Let the high school students translate each quadrilateral and graph the image on the grid. Label the quadrilateral after transformation.

Transformation: Any Two of Three

Two types of transformation have been performed to each figure. Middle school children should choose the correct transformations undergone.

Write the Rules

Identify the transformation undergone by the figure and write a rule to describe each of them.

Writing Coordinates: With Graph

Perform the required transformation for each figure and graph it. Also write the coordinates of the image obtained. Suitable for 8th graders.

Writing New Coordinates

The coordinates of the figure are given. Write down the coordinates of the vertices of the image after transformation.

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Unit 10: Transformations

About this unit.

In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations.

You will learn how to perform the transformations, and how to map one figure into another using these transformations.

Introduction to rigid transformations

• Rigid transformations intro (Opens a modal)
• Translations intro (Opens a modal)
• Rotations intro (Opens a modal)
• Identify transformations 4 questions Practice

Translations

• Translating shapes (Opens a modal)
• Determining translations (Opens a modal)
• Translation challenge problem (Opens a modal)
• Properties of translations (Opens a modal)
• Translations review (Opens a modal)
• Translate points 4 questions Practice
• Determine translations 4 questions Practice
• Translate shapes 4 questions Practice
• Rotating shapes (Opens a modal)
• Determining rotations (Opens a modal)
• Rotating shapes about the origin by multiples of 90° (Opens a modal)
• Rotations review (Opens a modal)
• Rotating shapes: center ≠ (0,0) (Opens a modal)
• Rotate points 4 questions Practice
• Determine rotations 4 questions Practice
• Rotate shapes 4 questions Practice
• Rotate shapes: center ≠ (0,0) 4 questions Practice

Reflections

• Reflecting shapes: diagonal line of reflection (Opens a modal)
• Determining reflections (advanced) (Opens a modal)
• Reflecting shapes (Opens a modal)
• Reflections review (Opens a modal)
• Reflect points 4 questions Practice
• Determine reflections 4 questions Practice
• Determine reflections (advanced) 4 questions Practice
• Reflect shapes 4 questions Practice
• Advanced reflections 4 questions Practice

Rigid transformations overview

• No videos or articles available in this lesson
• Find measures using rigid transformations 4 questions Practice
• Rigid transformations: preserved properties 4 questions Practice
• Mapping shapes 4 questions Practice
• Performing dilations (Opens a modal)
• Dilating shapes: shrinking by 1/2 (Opens a modal)
• Dilating shapes: expanding (Opens a modal)
• Dilate points 4 questions Practice
• Dilations: scale factor 4 questions Practice
• Dilations: center 4 questions Practice
• Dilate triangles 4 questions Practice
• Dilations and properties 4 questions Practice

Properties and definitions of transformations

• Precisely defining rotations (Opens a modal)
• Identifying type of transformation (Opens a modal)
• Sequences of transformations 4 questions Practice
• Defining transformations 4 questions Practice
• Intro to reflective symmetry (Opens a modal)
• Intro to rotational symmetry (Opens a modal)
• Finding a quadrilateral from its symmetries (Opens a modal)
• Finding a quadrilateral from its symmetries (example 2) (Opens a modal)
• Reflective symmetry of 2D shapes 4 questions Practice

Old transformations videos

• Performing translations (old) (Opens a modal)
• Performing rotations (old) (Opens a modal)
• Performing reflections: rectangle (old) (Opens a modal)
• Performing reflections: line (old) (Opens a modal)
• Determining translations (old) (Opens a modal)
• Rotation examples (old) (Opens a modal)
• Determining rotations (old) (Opens a modal)
• Dilating lines (Opens a modal)