Name ________________________________________ Date ___________________ Class __________________
LESSON
4-1
Reteach
Congruence and Transformations
TYPES OF TRANSFORMATIONS (centered at (0, 0))
Translation (slide): (x, y) → (x + a, y + b)
Reflection y-axis: (x, y) → (−x, y)
x-axis: (x, y) → (x, −y)
Rotation 90° clockwise: (x, y) → (y, −x)
Dilation: (x, y) → (kx, ky), k > 0
Rotation 90° counterclockwise: (x, y) → (−y, x)
Rotation 180°: (x, y) → (−x, −y)
Apply the transformation M to the polygon with the given vertices.
Identify and describe the transformation.
1. M: (x, y) → (x + 1, y − 2)
A(−1, −3), B(2, 2), C(−2, −1)
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2. M: (x, y) → (−x, −y)
P(0, 0), Q(1, 3), R(3, 3)
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4-6
Holt McDougal Geometry
Name ________________________________________ Date ___________________ Class __________________
LESSON
1-x
4-1
Reteach
Congruence and Transformations continued
An isometry is a transformation that preserves length, angle, and area. Because of these
properties, isometries produce congruent images. A rigid transformation is another name for
an isometry.
Dilations with scale factor k ≠ 1 are transformations that produce images that are not
congruent to their preimages.
Isometry
Image ≅ Preimage
translation
yes
yes
reflection
yes
yes
rotation
yes
yes
dilation
no
no
Transformation
Determine whether the polygons with the given vertices are
congruent.
3. E(−3, 1), F(−2, 4), G(0, 0)
4. R(−2, 4), S(0, 3), T(−3, −1)
H(1, −4), I(2, −1), J(4, −5)
U(2, 4), S(0, 3), V(3, −1)
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5. P(0, 0), Q(2, 2), R(−2, 1)
6. J(−2, 2), K(2, 1), L(1, 3)
P(0, 0), S(4, 4), T(−4, 2)
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P(−4, 4), Q(4, 2), R(2, 6)
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4-7
Holt McDougal Geometry
Answers for the chapter Triangle Congruence
4-1 CONGRUENCE AND
2.
TRANSFORMATIONS
Practice A
1. A′(7, 7), B′(−2, 1), C′(3, −10); translation
4 units right and 3 units down
2. P′(2, −6), Q′(8, 2), R′(−4, −2); dilation
about (0, 0) with a scale factor of 2;
3. E′(−2, −2), F′(3, −4), G′(0, −5); reflection
across the x-axis
y-axis
4. R′(4, 3), S′(−2, −3), T′(−2, v2); rotation
90° clockwise with center (0, 0)
; reflection in the
3.
5. Yes; this is a translation 1 unit left and 2
units up.
6. Yes; this is a translation 2 units right and
4 units down.
7. No; the triangles are not congruent
because triangle GHL can be mapped
onto triangle PQR by a dilation with a
scale factor of 2 and a center of (0, 0).
about (0, 0), 90° clockwise
; rotation
4.
8. Yes; this is a reflection across the y-axis.
9. A′(−6, 0), B′(−3, 7), C′(−1, 6); this is a
translation 2 units to the left and 2 units
up
10. X′(−1, −2), Y′(−4, −1), Z′(−3, 1); this is a
rotation 180° clockwise with center (0, 0)
; dilation with
scale factor 2 and center (0, 0)
Practice B
1.
units left and 3 units up
5.
Yes, the pentagons are congruent
because pentagon ABCDE can be
mapped to pentagon PQRST by a
translation: (x, y) → (x + 6, y + 2).
6.
No, the triangles are not congruent
because triangle JKL can be mapped to
triangle PQR by a dilation with scale
factor 2 and a center of (0, 0).
Practice C
; translation 2
1. A′(2, 12), B′(−7, 6), C′(−2, −2); translation
1 unit left and 2 units up
2. A′(0.5, −1.5), B′(2, 0.5), C′(−1, −0.5);
dilation about (0, 0) with a scale factor of
1
2
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Holt McDougal Geometry
3. Yes; this is a translation 3 units right and
4 units down.
5. No, the triangles are not congruent
because triangle PQR can be mapped to
triangle PST by a dilation with scale
factor 2 and a center of (0, 0).
4. Yes; this is a dilation with center (0, 0)
and scale factor 2.5
6. No, the triangles are not congruent
because triangle JKL can be mapped to
triangle PQR by a dilation with scale
factor 2 and a center of (0, 0).
5. The triangles are congruent because
triangle ABC can be mapped onto
triangle DEF by a translation 3 units right
and 4 units down: (x, y) → (x + 3, y − 4).
6. The quadrilaterals are congruent because
JKLM can be mapped onto QRST by a
rotation 90° clockwise: (x, y) → (y, −x).
Challenge
1. Sample answer:
7. A′(6, 2), B′(6, 6), C′(10, 2); this is a
reflection in the x-axis followed by a
translation 4 units to the right.
10. X′(−2, 4), Y′(−6, −2), Z′(−8, 2); this is a
dilation of scale factor 2 and center (0, 0).
Then a reflection in the y-axis.
2. Sample answer:
Reteach
1.
3. Sample answer:
; translation 1
unit right and 2 units down
2.
4. Sample answer:
; rotation 180°
3. Yes, the triangles are congruent because
triangle EFG can be mapped to triangle
HIJ by a translation: (x, y) → (x + 4,
y − 5).
5. Sample answer:
4. Yes, the triangles are congruent because
triangle RST can be mapped to pentagon
USV by a reflection in the y-axis.
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