Introduction

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Think It Through

Lesson 18 Understand Properties of Transformations

Lesson 18Understand Properties of Transformations

There are three ways to move a figure from one place to another. These movements are called transformations.

1. You can slide the figure. 2. You can reflect the figure. 3. You can rotate the figure.

A translation, or slide, moves every point of a figure the same distance and in the same direction. The figure moves left, right, up, or down. Here are some examples of translations.

Right

Up

Right and up

Think What is a translation?

Circle the words that describe how a figure moves in a translation.

What are different ways a figure can be moved?

M.8.16a

M.8.16b

M.8.16c

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A reflection, or flip, flips a figure over an imaginary line that can be horizontal, vertical, or slanted. The line is called a line of reflection. Imagine folding on the line of reflection. The original figure and its image would match exactly. Corresponding points on the original figure and its image are the same distance from the line of reflection. Here are some examples of reflections.

Horizontal line

Vertical line

Slanted line

A rotation, or turn, moves a figure around a fixed point. The fixed point is called the center of rotation. The figure turns in either a clockwise or counterclockwise direction. Here are some examples of rotations.

Clockwise

Counterclockwise

Reflect1 When a figure is translated, reflected, or rotated, what’s the same about the original

figure and its image? What’s different?

Think What is a reflection?

A 360° rotation turns a full circle.

Think What is a rotation?

Guided Instruction

Think About

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Lesson 18

Lesson 18 Understand Properties of Transformations

Exploring Properties of Transformations

The figure on the right is a transformation of the figure on the left. Each square represents 1 unit.

2 Is the transformation a translation, reflection, or rotation? Explain your reasoning.

3 How do the lengths of the sides in the original figure compare to the lengths of the sides in its image? How do you know?

4 Describe the angles in the original figure.

Describe the angles in the image.

5 Are any of the sides of the original figure parallel?

Are any of the sides parallel in the image?

Now try this problem.

6 Use a separate sheet of paper to trace the red figure shown. Cut out the figure. Use your cutout to decide how you could move the red figure to get each blue figure. Write the letter of the figure that matches the transformation and then complete the problem.

Translation Describe the move.

Reflection Draw the line of reflection.

Rotation Draw the point that is the center of rotation.

Let’s Explore the Idea Analyzing a figure and its image can help you understand the properties of transformations.

B

A

C

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Let’s Talk About It Solve the problems below as a group.

7 Look at your answers to problems 3–5 on the previous page. What properties of a figure’s lines and angles remain the same when you reflect the figure?

8 Trace the red hexagon at the top of the previous page. Cut out the figure. Use the cutout

and a separate sheet of paper to create translations and rotations of the figure. What

properties of a figure’s lines and angles remain the same when you translate or rotate the

figure?

Try It Another Way Work with your group to explore properties of a rotation using rulers and protractors.

The blue triangle is a rotation of the red triangle.

9 Measure the lengths of the sides of DABC to the nearest cm and measure the angles to the nearest degree.

AB 5 BC 5 AC 5

/A 5 /ABC 5 /C 5

10 Measure the lengths of the sides of DXBY to the nearest cm and measure the angles to the nearest degree.

XB 5 BY 5 YX 5

/X 5 /XBY 5 /Y 5

11 Compare the triangles. What can you say about the lengths of the sides and the measures of the angles of the two triangles?

A

C B

X

Y

Guided Practice

Connect

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Lesson 18

Lesson 18 Understand Properties of Transformations

Exploring Properties of Transformations

Talk through these problems as a class, then write your answers below.

12 Explain Tell whether the blue figure appears to be a reflection of the red figure. Explain why or why not.

a.

b.

c.

13 Draw Imagine spinning this flag clockwise around in a circle about point O. Draw a 90° rotation, a 180° rotation, and a 270° rotation. Use a different color for each rotation.

A

O

B

C

14 Analyze Refer back to the rotated flag in Problem 13. What is always true about ··· OA and ··· OB in each of the images of rotation?

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Independent Practice

Apply

Lesson 18

Lesson 18 Understand Properties of Transformations

Exploring Properties of Transformations

15 Put It Together Use what you have learned to complete this task.

Alan created each design by moving the same triangle in different ways and tracing it after each move.

I

II

Part A Identify the type of transformation that created the designs shown above.

I II

Part B Choose a different figure and make your own design on the grid below using one type of transformation. Describe how you moved the figure to make your design.