Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry 171 172 amp Symmetry Warm Up Determine the coordinates of the image of P 4 7 under each transformation ID: 573060
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Slide1
Compositions of Transformations
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
17.1
17.2
&
SymmetrySlide2
Warm Up
Determine the coordinates of the image of P(4, –7) under each transformation.
1. a translation 3 units left and 1 unit up
2. a rotation of 90° about the origin (1, –6)(7, 4)
3. a reflection across the y-axis (–4, –7)17.1Slide3
A
composition of transformations is one transformation followed by another. For example, a glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector.
17.1Slide4
The glide reflection that maps
∆JKL to ∆J’K’L’ is the composition of a translation along followed by a reflection across line
l.
17.1Slide5
The image after each transformation is congruent to the previous image. By the Transitive Property of Congruence, the final image is congruent to the preimage. This leads to the following theorem.
17.1Slide6
Example
1: Drawing Compositions of
Isometries
Draw the result of the composition of isometries.∆KLM has vertices K(4, –1), L(5, –2), and M(1, –4). Rotate ∆KLM 180° about the origin and then reflect it across the y-axis.
K
L
M
17.1Slide7
Example
1 Continued
Step 1 The rotational image of (
x, y) is (–x, –y). K(4, –1) K’(–4, 1
), L(5, –2) L’(–5, 2), and M(1, –4) M’(–1, 4).
Step 2 The reflection image of (x, y) is (–x, y). K’(
–4, 1) K”(4, 1),
L’(–5, 2) L”(5, 2
), and M’(–1, 4)
M”(1, 4).
Step 3 Graph the image and preimages.
K
L
M
M’
K’
L’
L”
M”
K”
17.1Slide8
17.1Slide9
Example
2:
Art Application
Sean reflects a design across line p and then reflects the image across line q. Describe a single transformation that moves the design from the original position to the final position.By Theorem 12-4-2, the composition of two reflections across parallel lines is equivalent to a translation perpendicular to the lines. By Theorem 12-4-2, the translation vector is 2(5 cm) = 10 cm to the right.17.1Slide10
17.1Slide11
Diatoms are microscopic algae that are found in aquatic environments. Scientists use a system that was developed in the 1970s to classify diatoms based on their
symmetry.
A figure has symmetry if there is a transformation of the figure such that the image coincides with the preimage.17.2Slide12
17.2Slide13
Example
3: Identifying line of symmetry
Yes; four lines of symmetry
Tell whether the figure has line symmetry. If so, copy the shape and draw all lines of symmetry.17.2Slide14
Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry.
Example 4
yes; two lines of symmetry
a.
b.
c.yes; one line of symmetry
yes; one line of symmetry
17.2Slide15
17.2Slide16
The
angle of rotational symmetry is the smallest angle through which a figure can be rotated to coincide with itself. The number of times the figure coincides with itself as it rotates through 360° is called the
order of the rotational symmetry.
Angle of rotational symmetry: 90° Order: 417.2Slide17
Example
5: Identifying Rotational Symmetry
Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry.
no rotational symmetry
yes; 180°; order: 2yes; 90°;order: 4
A.B.C.17.2
D.
yes; 120°; order: 3Slide18
Example
6: Design Application
Describe the symmetry of each icon. Copy each shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order.
No line symmetry; rotational symmetry; angle of rotational symmetry: 180°; order: 217.2Slide19
Example
7: Design Application
Line symmetry and rotational symmetry;
angle of rotational symmetry: 90°; order: 4Describe the symmetry of each icon. Copy each shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order.
17.2Slide20
Example 8
Describe the symmetry of each diatom. Copy the shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order.
line symmetry and rotational symmetry; 72°; order: 5
a.
b.line symmetry and rotational symmetry; 51.4°; order: 7
17.2