Slide, Flip, Turn! - Amazon Web Services · 121 2013 COURSE 3 slide, flip, Turn! e ver Y Gr A ph...

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everY GrAph Tells A sTorY • slide, flip, Turn! • A pArk rAnGer’s work is never done

{ Student Text Reference Page 429 }

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8.1 Translations, Rotations, and Reflections of Triangles • 429

Learning GoalsIn this lesson, you will:

Translate triangles in a coordinate plane.

Rotate triangles in a coordinate plane.

Reflect triangles in a coordinate plane.

Slide, Flip, Turn!Translations, Rotations, and Reflections of Triangles

When you look at the night sky, you see bright stars and dim stars. But are the

dimmer stars farther away from us or just less bright? Astronomers use a variety

of methods to measure the universe, but at the end of the 1980s, they made vast

improvements in the accuracy of these measurements.

In 1989, the Hipparcos satellite was launched by the European Space Agency.

Among other advantages, this satellite was not affected by Earth’s atmosphere

and could view the entire “sky,” so it could provide more accurate measurements

of distances. In 1997, the Hipparcos Catalogue was published, which contained

high-precision distance information for more than 100,000 stars!

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Copyright © 2013 Carnegie Learning, Inc.


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430 • Chapter 8 Congruence of Triangles

Problem 1 Translating Triangles in a Coordinate Plane

You have studied translations, rotations, and reflections of various geometric figures.

In this lesson, you will explore, compare, and generalize the characteristics of triangles as

you translate, rotate, and reflect them in a coordinate plane.

Consider the point (x, y) located anywhere in the first quadrant of the coordinate plane.

x

2

2 4 6 8 10

4

6

8

–2–2–4–6–8–10

–4

–6

–8

y

10

–10

(x, y)

0

1. Translate the point (x, y) according to the descriptions in the table shown. Plot the

point, and then record the coordinates of the translated points in terms of x and y.

Translation Point (x, y) located in Q1

3 units to the left

3 units down

3 units to the right

3 units up

The ordered pair (x , y )

represents any point that is located in the

first quadrant.

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2. Describe the translation in terms of x and y that would move any point (x, y) into:

a. Quadrant II

b. Quadrant III

c. Quadrant IV

3. Graph triangle ABC by plotting the points A(23, 4), B(26, 1),

and C(24, 9).

x86

2

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

Use the table to record the coordinates of the vertices of each triangle.

a. Translate triangle ABC 5 units to the right

to form triangle A9B9C9. List the coordinates

of points A9, B9, and C9. Then graph

triangle A9B9C9.

b. Translate triangle ABC 8 units down to form triangle A0B0C0.

List the coordinates of points A0, B0, and C0. Then graph

triangle A0B0C0.

Can you translate a point from QI to QIII in one move?

Triangle ABC is located in Quadrant

II. Do you think any of these translations will change

the quadrant location of the triangle?


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{ Student Text Reference 32 }432 • Chapter 8 Congruence of Triangles

Original TriangleTriangle Translated 5 units to the Right

Triangle Translated 8 units Down

△ABC △A9B9C9 △A0B0C0

A (23, 4)

B (26, 1)

C (24, 9)

Let’s consider the vertices of a different triangle and translations without graphing.

4. The vertices of triangle DEF are D(27, 10), E(25, 5), and F(28, 1).

a. If triangle DEF is translated to the right 12 units, what are the coordinates of the

vertices of the image? Name the triangle.

b. How did you determine the coordinates of the image without graphing

the triangle?

c. If triangle DEF is translated up 9 units, what are the

coordinates of the vertices of the image? Name

the triangle.

d. How did you determine the coordinates of the image without

graphing the triangle?

Think about which values of

the ordered pairs are changing.

Create a table if it helps you organize the

vertices.

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433

Problem 2 Rotating Triangles in a Coordinate Plane

1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane.

x

2

4

6

8

–2

–4

–6

–8

y

10

–10

20

4 6 8 10–2–4–6–8–10 0

Use the table to record the coordinates of each point.

a. Using the origin (0, 0) as the point of rotation,

rotate point (x, y) 90° counterclockwise about the

origin and graph the rotated point on the

coordinate plane. What are the new coordinates

of the rotated point in terms of x and y?

b. Using the origin (0, 0) as the point of rotation,

rotate point (x, y) 180° counterclockwise about the

origin and graph the rotated point on the coordinate plane. What

are the new coordinates of the rotated point in terms of x and y?

Original PointRotation About

the Origin90° Counterclockwise

Rotation About the Origin

180° Counterclockwise

(x, y)

If your point was at (5, 0), and you

rotated it 90°, where would it end up? What about

if it was at (5, 1)?

Use your straightedge

when drawing the 90° angle.

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434 • Chapter 8 Congruence of Triangles

2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9).

x86

2

4

8

10–2–2

42–4

–4

–6

–8

–8

–10

y

10

–10

6

–60


a. Using the origin (0, 0) as the point of rotation, rotate triangle ABC 90°

counterclockwise about the origin to form triangle A9B9C9. Graph the triangle

and then list the coordinates of the rotated triangle.

b. Using the origin (0, 0) as the point of rotation, rotate triangle ABC 180°

counterclockwise about the origin to form triangle A0B0C0. Graph the triangle and

then list the coordinates of the rotated triangle.

Original TriangleRotation About

the Origin90° Counterclockwise

Rotation About the Origin

180° Counterclockwise


A (3, 4)

B (6, 1)

C (4, 9)

Think about your answers from Question 1 as you

rotate the triangle.

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Let’s consider a different triangle and rotations without graphing.


a. If triangle DEF is rotated 90° counterclockwise, what are the coordinates of the

vertices of the image? Name the rotated triangle.


the triangle?

c. If triangle DEF is rotated 180° counterclockwise, what are the coordinates of the

vertices of the image? Name the rotated triangle.

d. How did you determine the coordinates of the image without graphing

the triangle?

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{ Student Text Reference Page 436 }436 • Chapter 8 Congruence of Triangles

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Problem 3 Reflecting Triangles in a Coordinate Plane

1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane.

x

2

2 4 6 8 10

4

6

8

–2

–2–4–6–8–10

–4

–6

–8

y

10

–10

0

Use the table to record the coordinates of each point.

a. Reflect and graph the point (x, y) over the x-axis on the coordinate plane. What are

the new coordinates of the reflected point in terms of x and y?

b. Reflect and graph the point (x, y) over the y-axis on the coordinate plane. What are

the new coordinates of the reflected point in terms of x and y?

Original PointReflection Over the

x-axisReflection Over the

y-axis

(x, y)

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2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9).

x86

2

4

8

10–2–2

42–4

–4

–6

–8

–8

–10

y

10

–10

6

–60


a. Reflect triangle ABC over the x-axis to form triangle A9B9C9. Graph the triangle and

then list the coordinates of the reflected triangle.

b. Reflect triangle ABC over the y-axis to form triangle A0B0C0. Graph the triangle and

then list the coordinates of the reflected triangle.

Original TriangleTriangle Reflected

Over the x-axisTriangle Reflected

Over the y-axis


A (3, 4)

B (6, 1)

C (4, 9)

Do you see any patterns?

8.1 Translations, Rotations, and Reflections of Triangles • 437slide, flip, Turn!



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Let’s consider a different triangle and reflections without graphing.


a. If triangle DEF is reflected over the x-axis, what are the coordinates of the vertices

of the image? Name the triangle.


the triangle?

c. If triangle DEF is reflected over the y-axis, what are the coordinates of the vertices

of the image? Name the triangle.

d. How did you determine the coordinates of the image without graphing

the triangle?

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Talk the Talk

1. The vertices of triangle PQR are P(4, 3), Q(22, 2), and R(0, 0). Describe the translation

used to form each triangle. Explain your reasoning.

a. P9(0, 3), Q9(26, 2), and R9(24, 0)

b. P0(4, 5.5), Q0(22, 4.5), and R0(0, 2.5)

2. The vertices of triangle JME are J(1, 3), M(6, 5), and E(8, 1). Describe the rotation used

to form each triangle. Explain your reasoning.

a. J9(23, 1), M9(25, 6), and E9(21, 8)

b. J0(21, 23), M0(26, 25), and E0(28, 21)

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3. The vertices of triangle NRT are N(12, 4), R(14, 1), and T(20, 9). Describe the reflection

used to form each triangle. Explain your reasoning.

a. N9(212, 4), R9(214, 1), and T9(220, 9)

b. N0(12, 24), R0(14, 21), and T0(20, 29)

4. Are all the images that result from a translation, rotation, or reflection (always,

sometimes, or never) congruent to the original figure?

Be prepared to share your solutions and methods.

Remember, congruence

preserves size and shape.

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