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Lesson 3.4.1Lesson 3.4.1

ReflectionsReflections

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Lesson

3.4.1ReflectionsReflections

California Standard:Measurement and Geometry 3.2Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.

What it means for you:You’ll learn what it means to reflect a shape. You’ll also see how to draw and describe reflections.

Key words:• reflection• image• flip• prime• coordinates• x-axis/y-axis

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ReflectionsReflectionsLesson

3.4.1

The next few Lessons are about transformations.

The first type of transformation you’re going to meet is reflection.

For example, it could be flipping, stretching, moving, enlarging, or shrinking the shape.

A transformation is a way of changing a shape.

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ReflectionsReflections

A Reflection Flips a Figure Across a Line

Lesson

3.4.1

A reflection takes a shape and makes a mirror image of it on the other side of a given line.

The whole reflected triangle A'B'C' is called the image of ABC.

The reflections of points A, B, and C are labeled A', B', and C'.

Here triangle ABC has been reflected or “flipped” across the line of reflection.

A

B

C

A'

C'

B'

A' is read as “A prime.”

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ReflectionsReflections

Example 1

Solution follows…

Lesson

3.4.1

Reflect triangle DEF across the y-axis.

–8 –6 –4 –2 2 4 6 80

2

4

6y

0 x

DE

F

Solution

D'

7 units 7 units

Step 1: Pick a point to reflect — point D.

Move across the y-axis and find the point 7 units away on the other side. This is where you plot the point D'.

Point D is 7 units away from the y-axis.

Solution continues…

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ReflectionsReflections

Example 1

Lesson

3.4.1

Reflect triangle DEF across the y-axis.

–8 –6 –4 –2 2 4 6 80

2

4

6y

0 x

DE

F

Solution (continued)

D'

Step 2: Repeat step 1 for points E and F.

Step 3: Join the points to complete triangle D'E'F'.

E'

F'

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ReflectionsReflections

Guided Practice

Solution follows…

Lesson

3.4.1

In Exercises 1–2, copy each shape onto a set of axes, then draw its reflections across the y-axis and the x-axis.

1. 2.

–6 –4 –2 2 4 60

2

4

6y

0x

–6

–4

–2–6 –4 –2 2 4 60

2

4

6y

0x

–6

–4

–2

R

ST

K L

MN

R'

S'T'

T'' S''

R''

L' K'

N'M'

N''

M''

K'' L''

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ReflectionsReflections

Guided Practice

Solution follows…

Lesson

3.4.1

In Exercises 3–4, copy each shape onto a set of axes, then draw its reflections across the y-axis and the x-axis.

3. 4.

–6 –4 –2 2 4 60

2

4

6y

0x

–6

–4

–2–6 –4 –2 2 4 60

2

4

6y

0x

–6

–4

–2G H

I

ZW

X

Y

I''

G'' H''

I'

G'H'

W''

X''

Y''

Z''

W'

X'

Y'

Z'

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ReflectionsReflections

Reflections Change Coordinate Signs

Lesson

3.4.1

A reflection across the x-axis changes (x, y) to (x, –y).A reflection across the y-axis changes (x, y) to (–x, y).

To see this, look again at the reflection from Example 1. The coordinates of the corners of the triangles are shown.

–8 –6 –4 –2 2 4 6 80

2

4

6y

0 x

DE

F

D'E'

F'

D (–7, 4)E (–2, 5)F (–3, 2)

D' (7, 4)E' (2, 5)F' (3, 2)

When DEF is reflected across the y-axis, the y-coordinate stays the same and the x-coordinate changes from negative to positive.

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ReflectionsReflections

If you reflect DEF across the x-axis, the x-coordinate stays the same and the y-coordinate changes from positive to negative.

Lesson

3.4.1

D (–7, 4)E (–2, 5)F (–3, 2)

D'' (–7, –4)E'' (–2, –5)F'' (–3, –2)

When you’re drawing more than one image, the first should be called A' B' C', the second is A'' B'' C'' and so on.

–8 –6 –4 –2 0

2

4

6

0x

D E

F

D"E"

F"

y

–2

–4

–6

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ReflectionsReflections

Guided Practice

Solution follows…

Lesson

3.4.1

In Exercises 5–8, give the coordinates of the image produced.

5. A: (5, 2), (4, 7), (6, 1). Triangle A is reflected over the x-axis.

6. B: (9, 9), (–4, 8), (–2, 6). Triangle B is reflected over the y-axis.

7. C: (–2, 10), (2, 10), (5, 5), (0, –3), (–5, 5). Pentagon C is reflected over the x-axis.

8. Pentagon C from Exercise 7 is reflected over the y-axis.

(5, –2), (4, –7), (6, –1)

(–9, 9), (4, 8), (2, 6)

(–2, –10), (2, –10), (5, –5), (0, 3), (–5, –5)

(2, 10), (–2, 10), (–5, 5), (0, –3), (5, 5)

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ReflectionsReflections

Guided Practice

Solution follows…

Lesson

3.4.1

Exercises 9–11 give the coordinates of the corners of a figure and its reflected image. Describe each reflection in words.

9. D: (5, 2), (6, 3), (8, 1), (4, 1); D': (5, –2), (6, –3), (8, –1), (4, –1)

10. E: (–6, –1), (–3, –6), (–9, –4); E': (6, –1), (3, –6), (9, –4)

11. F: (0, 0), (0, 5), (3, 3); F': (0, 0), (0, 5), (–3, 3)

Reflection over x-axis

Reflection over y-axis

Reflection over y-axis

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ReflectionsReflections

Independent Practice

Solution follows…

Lesson

3.4.1

Copy the grid and figures shown below, then draw the reflections described in Exercises 1–3.

1. Reflect A across the x-axis. Label the image A'.

2. Reflect A across the y-axis. Label the image A''.

3. Reflect B across the x-axis. Label the image B'.

–10 –5

–5

5 100

5

10y

0x

–10

A

B

A'

A''

B'

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ReflectionsReflections

Independent Practice

Solution follows…

Lesson

3.4.1

Copy the grid and figures shown below, then draw the reflections described in Exercises 4–6.

4. Reflect B across the y-axis. Label the image B''.

5. Reflect C across the x-axis. Label the image C'.

6. Reflect C across the y-axis. Label the image C''.

–10 –5

–5

5 100

5

10y

0x

–10B

B''

C

C'

C''

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ReflectionsReflections

Independent Practice

Solution follows…

Lesson

3.4.1

In Exercises 7–8, copy the figures onto graph paper and reflect each one over the line of reflection shown.

7. 8.

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ReflectionsReflections

Independent Practice

Solution follows…

Lesson

3.4.1

In Exercise 9, copy the figure onto graph paper and reflect it over the line of reflection shown.

9.

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ReflectionsReflections

Round UpRound Up

Lesson

3.4.1

Don’t forget that a reflection makes a back-to-front image — like the image you see when you look in a mirror. Unless the original is symmetrical, the image shouldn’t be the same way around as the original. If it is the same way around, that’s a translation, not a reflection.

You’ll learn about translations in the next Lesson.