Slide, Flip, Turn!
Slide, Flip, Turn! Course 3
Translations, Rotations, and Reflections of Triangles
Learning Goals In this lesson, you will:
W
hen 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
© 2011 Copyright © 2013 Carnegie Learning, Inc.
Carnegie Learning
high-precision distance information for more than 100,000 stars!
{ Student Text Reference Page 429 }
8.1
Translations, Rotations, and Reflections of Triangles
•
429
Slide, Flip, Turn!
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
Translate triangles in a coordinate plane. Rotate triangles in a coordinate plane. Reflect triangles in a coordinate plane.
119
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. y
The ordered pair (x , y ) represents any point that is located in the first quadrant.
10 8 6 (x, y)
4 2 –10
–8
–6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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 to the right 3 units up
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Congruence of Triangles
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3 units down
2. Describe the translation in terms of x and y that would move any point (x, y) into: a. Quadrant II
Course 3
Can you translate a point from QI to QIII in one move? b. Quadrant III
3. Graph triangle ABC by plotting the points A(23, 4), B(26, 1), and C(24, 9). y 10 8 6 4 2 –10 –8
–6
–4
0
–2
2
4
6
8
–2 –4 –6 –8 –10
10 x
Triangle ABC is located in Quadrant II. Do you think any of these translations will change the quadrant location of the triangle?
Use the table to record the coordinates of the vertices of each triangle. a. Translate triangle ABC 5 units to the right
Copyright © 2013 Carnegie Learning, Inc.
to form triangle A9B9C9. List the coordinates
{ Student Text Reference Page 431 }
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.
8.1
Translations, Rotations, and Reflections of Triangles
•
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
c. Quadrant IV
431 Slide, Flip, Turn! 121
Original Triangle
Triangle 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
Think about which values of the ordered pairs are changing.
the triangle?
c. If triangle DEF is translated up 9 units, what are the coordinates of the vertices of the image? Name the triangle.
Create a table if it helps you organize the vertices.
d. How did you determine the coordinates of the image without graphing the triangle? Copyright © 2013 Carnegie Learning, Inc.
122
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•
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Congruence of Triangles
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Problem 2
Rotating Triangles in a Coordinate Plane
1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane.
Course 3
y 10 8
Use your straightedge when drawing the 90° angle.
6 4 2 0 –6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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?
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)?
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
Original Point
Rotation About the Origin 90° Counterclockwise
Rotation About the Origin 180° Counterclockwise
(x, y)
Copyright © 2013 Carnegie Learning, Inc.
© 2011 Carnegie Learning
are the new coordinates of the rotated point in terms of x and y?
{ Student Text Reference Page 433 }
Slide, Flip, Turn!
433
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
–10 –8
123
2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9). y 10 8 6 4 2 –10 –8
Think about your answers from Question 1 as you rotate the triangle.
–6
–4
–2 –2
0
2
4
6
8
10 x
–4 –6 –8 –10
Use the table to record the coordinates of the vertices of each triangle. 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 Triangle
Rotation About the Origin 90° Counterclockwise
Rotation About the Origin 180° Counterclockwise
△ABC
△A9B9C9
△A0B0C0
B (6, 1) C (4, 9)
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© 2011 Carnegie Learning
A (3, 4)
Let’s consider a different triangle and rotations without graphing. 3. The vertices of triangle DEF are D(27, 10), E(25, 5), and F(21, 28). a. If triangle DEF is rotated 90° counterclockwise, what are the coordinates of the vertices of the image? Name the rotated triangle.
Course 3
b. How did you determine the coordinates of the image without graphing the triangle?
vertices of the image? Name the rotated triangle.
d. How did you determine the coordinates of the image without graphing
© 2011 Copyright © 2013 Carnegie Learning, Inc.
Carnegie Learning
the triangle?
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8.1
Translations, Rotations, and Reflections of Triangles
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
c. If triangle DEF is rotated 180° counterclockwise, what are the coordinates of the
• Slide,435 Flip, Turn! 125
Problem 3
Reflecting Triangles in a Coordinate Plane
1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane. y 10 8 6 4 2 –10 –8
–6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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 Point
Reflection Over the x-axis
Reflection Over the y-axis
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Congruence of Triangles
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© 2011 Carnegie Learning
(x, y)
2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9). y 10 8
Course 3
6 4 2 –10 –8
–6
–4
–2 –2
0
2
4
6
8
10 x
–4
–8 –10
Use the table to record the coordinates of the vertices of each triangle. 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 Triangle
Triangle Reflected Over the x-axis
Triangle Reflected Over the y-axis
△ABC
△A9B9C9
△A0B0C0
A (3, 4) B (6, 1)
Do you see any patterns? Copyright © 2013 Carnegie Learning, Inc.
© 2011 Carnegie Learning
C (4, 9)
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8.1
Translations, Rotations, and Reflections of Triangles
•
437
Slide, Flip, Turn!
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
–6
127
Let’s consider a different triangle and reflections without graphing. 3. The vertices of triangle DEF are D(27, 10), E(25, 5), and F(21, 28). a. If triangle DEF is reflected over the x-axis, 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 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
Copyright © 2013 Carnegie Learning, Inc.
128
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•
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Congruence of Triangles
{ Student Text Reference Page 438 }
© 2011 Carnegie Learning
the triangle?
Talk the Talk 1. The vertices of triangle PQR are P(4, 3), Q(22, 2), and R(0, 0). Describe the translation
Course 3
used to form each triangle. Explain your reasoning. a. P9(0, 3), Q9(26, 2), and R9(24, 0)
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)
© 2011 Copyright © 2013 Carnegie Learning, Inc. Carnegie
Learning
b. J0(21, 23), M0(26, 25), and E0(28, 21)
{ Student Text Reference Page 439 }
8.1
Translations, Rotations, and Reflections of Triangles
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
b. P0(4, 5.5), Q0(22, 4.5), and R0(0, 2.5)
• Slide,439 Flip, Turn! 129
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?
130
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440
•
Chapter 8
Congruence of Triangles
Copyright © 2013 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
{ Student Text Reference Page 440 }
© 2011 Carnegie Learning
Remember, congruence preserves size and shape.
Translations, Rotations, and Reflections of Triangles
Learning Goals In this lesson, you will:
W
hen 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
© 2011 Copyright © 2013 Carnegie Learning, Inc.
Carnegie Learning
high-precision distance information for more than 100,000 stars!
{ Student Text Reference Page 429 }
8.1
Translations, Rotations, and Reflections of Triangles
•
429
Slide, Flip, Turn!
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
Translate triangles in a coordinate plane. Rotate triangles in a coordinate plane. Reflect triangles in a coordinate plane.
119
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. y
The ordered pair (x , y ) represents any point that is located in the first quadrant.
10 8 6 (x, y)
4 2 –10
–8
–6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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 to the right 3 units up
Copyright © 2013 Carnegie Learning, Inc.
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Congruence of Triangles
{ Student Text Reference Page 430 }
© 2011 Carnegie Learning
3 units down
2. Describe the translation in terms of x and y that would move any point (x, y) into: a. Quadrant II
Course 3
Can you translate a point from QI to QIII in one move? b. Quadrant III
3. Graph triangle ABC by plotting the points A(23, 4), B(26, 1), and C(24, 9). y 10 8 6 4 2 –10 –8
–6
–4
0
–2
2
4
6
8
–2 –4 –6 –8 –10
10 x
Triangle ABC is located in Quadrant II. Do you think any of these translations will change the quadrant location of the triangle?
Use the table to record the coordinates of the vertices of each triangle. a. Translate triangle ABC 5 units to the right
Copyright © 2013 Carnegie Learning, Inc.
to form triangle A9B9C9. List the coordinates
{ Student Text Reference Page 431 }
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.
8.1
Translations, Rotations, and Reflections of Triangles
•
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
c. Quadrant IV
431 Slide, Flip, Turn! 121
Original Triangle
Triangle 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
Think about which values of the ordered pairs are changing.
the triangle?
c. If triangle DEF is translated up 9 units, what are the coordinates of the vertices of the image? Name the triangle.
Create a table if it helps you organize the vertices.
d. How did you determine the coordinates of the image without graphing the triangle? Copyright © 2013 Carnegie Learning, Inc.
122
432
Course 3
•
Chapter 8
Congruence of Triangles
{ Student Text Reference Page 432 }
Problem 2
Rotating Triangles in a Coordinate Plane
1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane.
Course 3
y 10 8
Use your straightedge when drawing the 90° angle.
6 4 2 0 –6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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?
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)?
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
Original Point
Rotation About the Origin 90° Counterclockwise
Rotation About the Origin 180° Counterclockwise
(x, y)
Copyright © 2013 Carnegie Learning, Inc.
© 2011 Carnegie Learning
are the new coordinates of the rotated point in terms of x and y?
{ Student Text Reference Page 433 }
Slide, Flip, Turn!
433
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
–10 –8
123
2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9). y 10 8 6 4 2 –10 –8
Think about your answers from Question 1 as you rotate the triangle.
–6
–4
–2 –2
0
2
4
6
8
10 x
–4 –6 –8 –10
Use the table to record the coordinates of the vertices of each triangle. 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 Triangle
Rotation About the Origin 90° Counterclockwise
Rotation About the Origin 180° Counterclockwise
△ABC
△A9B9C9
△A0B0C0
B (6, 1) C (4, 9)
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{ Student Text Reference Page 434 }
© 2011 Carnegie Learning
A (3, 4)
Let’s consider a different triangle and rotations without graphing. 3. The vertices of triangle DEF are D(27, 10), E(25, 5), and F(21, 28). a. If triangle DEF is rotated 90° counterclockwise, what are the coordinates of the vertices of the image? Name the rotated triangle.
Course 3
b. How did you determine the coordinates of the image without graphing the triangle?
vertices of the image? Name the rotated triangle.
d. How did you determine the coordinates of the image without graphing
© 2011 Copyright © 2013 Carnegie Learning, Inc.
Carnegie Learning
the triangle?
{ Student Text Reference Page 435 }
8.1
Translations, Rotations, and Reflections of Triangles
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
c. If triangle DEF is rotated 180° counterclockwise, what are the coordinates of the
• Slide,435 Flip, Turn! 125
Problem 3
Reflecting Triangles in a Coordinate Plane
1. Graph the point (x, y) anywhere in the first quadrant of the coordinate plane. y 10 8 6 4 2 –10 –8
–6
–4
0
–2
2
4
6
8
10 x
–2 –4 –6 –8 –10
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 Point
Reflection Over the x-axis
Reflection Over the y-axis
Copyright © 2013 Carnegie Learning, Inc.
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Congruence of Triangles
{ Student Text Reference Page 436 }
© 2011 Carnegie Learning
(x, y)
2. Graph triangle ABC by plotting the points A(3, 4), B(6, 1), and C(4, 9). y 10 8
Course 3
6 4 2 –10 –8
–6
–4
–2 –2
0
2
4
6
8
10 x
–4
–8 –10
Use the table to record the coordinates of the vertices of each triangle. 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 Triangle
Triangle Reflected Over the x-axis
Triangle Reflected Over the y-axis
△ABC
△A9B9C9
△A0B0C0
A (3, 4) B (6, 1)
Do you see any patterns? Copyright © 2013 Carnegie Learning, Inc.
© 2011 Carnegie Learning
C (4, 9)
{ Student Text Reference Page 437 }
8.1
Translations, Rotations, and Reflections of Triangles
•
437
Slide, Flip, Turn!
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
–6
127
Let’s consider a different triangle and reflections without graphing. 3. The vertices of triangle DEF are D(27, 10), E(25, 5), and F(21, 28). a. If triangle DEF is reflected over the x-axis, 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 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
Copyright © 2013 Carnegie Learning, Inc.
128
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•
Chapter 8
Congruence of Triangles
{ Student Text Reference Page 438 }
© 2011 Carnegie Learning
the triangle?
Talk the Talk 1. The vertices of triangle PQR are P(4, 3), Q(22, 2), and R(0, 0). Describe the translation
Course 3
used to form each triangle. Explain your reasoning. a. P9(0, 3), Q9(26, 2), and R9(24, 0)
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)
© 2011 Copyright © 2013 Carnegie Learning, Inc. Carnegie
Learning
b. J0(21, 23), M0(26, 25), and E0(28, 21)
{ Student Text Reference Page 439 }
8.1
Translations, Rotations, and Reflections of Triangles
Every Graph Tells a story • Slide, Flip, Turn! • A Park Ranger’s Work is Never Done
b. P0(4, 5.5), Q0(22, 4.5), and R0(0, 2.5)
• Slide,439 Flip, Turn! 129
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?
130
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440
•
Chapter 8
Congruence of Triangles
Copyright © 2013 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
{ Student Text Reference Page 440 }
© 2011 Carnegie Learning
Remember, congruence preserves size and shape.
