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slide, flip, Turn!
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!
120 Course 3
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{ Student Text Reference Page 430 }
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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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{ Student Text Reference Page 431 }
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?
8.1 Translations, Rotations, and Reflections of Triangles • 431
122 Course 3
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{ Student Text Reference Page 432 }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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{ Student Text Reference Page 433 }
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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.
124 Course 3
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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
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 TriangleRotation About
the Origin90° Counterclockwise
Rotation About the Origin
180° Counterclockwise
△ABC △A9B9C9 △A0B0C0
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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{ Student Text Reference Page 435 }
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8.1 Translations, Rotations, and Reflections of Triangles • 435
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.
b. How did you determine the coordinates of the image without graphing
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?
126 Course 3
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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
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 TriangleTriangle Reflected
Over the x-axisTriangle Reflected
Over the y-axis
△ABC △A9B9C9 △A0B0C0
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!
everY GrAph Tells A sTorY • slide, flip, Turn! • A pArk rAnGer’s work is never done
{ Student Text Reference Page 437 }
128 Course 3
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{ Student Text Reference Page 438 }438 • Chapter 8 Congruence of Triangles
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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
the triangle?
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{ Student Text Reference Page 439 }
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8.1 Translations, Rotations, and Reflections of Triangles • 439
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)
130 Course 3
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{ Student Text Reference Page 440 }440 • Chapter 8 Congruence of Triangles
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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.