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Holt Geometry
12-5 Symmetry
COMPOSITIONS OF TRANSFORMATIONS
• I can perform a composition of two or more transformations.
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Holt Geometry
12-5 Symmetry
How are transformations used?
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Holt Geometry
12-5 Symmetry
Compositions of TransformationsOne transformation followed by another.
You use the image of the first one as the preimage of the next.
ex: Reflect across the y-axis, then rotate 90°
****ORDER MATTERS******
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Holt Geometry
12-5 Symmetry
Example 1: Drawing Compositions of Isometries
∆ABC has vertices A(2, 6), B(7, 3), and C(3, –2).
Reflect across the x-axis
Rotate 90° A
B
C
A’
B’
C’A’’
B’’
C’’A’(2, -6)
B’(7, -3)
C’(3, 2)
A’’(6, 2)
B’’(3, 7)
C’’(-2, 3)
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Holt Geometry
12-5 Symmetry
Example 2: Drawing Compositions of Isometries
∆ABC has vertices A(2, 6), B(7, 3), and C(3, –2).
Translate <3, 3>
Reflect across y = x A
B
C
A’
B’
C’
A’’
B’’
C’’A’(5, 9)
B’(10, 6)
C’(6, 1)
A’’(9, 5)
B’’(6, 10)
C’’(1, 6)
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Holt Geometry
12-5 Symmetry
Example 3: Drawing Compositions of Isometries
∆ABC has vertices A(2, 6), B(7, 3), and C(3, –2).
Translate <2, 4>
Reflect across x-axis A
B
C
A’
B’
C’
A’’
B’’
C’’
A’(0, 10)
B’(5, 7)
C’(1, 2)
A’’(0, -10)
B’’(5, -7)
C’’(1, -2)
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Holt Geometry
12-5 Symmetry
Example 4: Drawing Compositions of Isometries
∆ABC has vertices A(2, 6), B(7, 3), and C(3, –2).
Translate <1, -3>
Reflect across y-axis A
B
C
A’
B’
C’
A’’
B’’
C’’
A’(3, 3)
B’(8, 0)
C’(4, -5)
A’’(-3, 3)
B’’(-8, 0)
C’’(-4, -5)
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Holt Geometry
12-5 Symmetry
Brain Stretch
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Holt Geometry
12-5 Symmetry12-5 Symmetry
Holt Geometry
I CANI CAN
•Identify line symmetry, rotational symmetry, and translational symmetry
•Name the pre-image and image points of a transformation
•Draw a line of symmetry for a given figure.
●Find the equation of a line of symmetry
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Holt Geometry
12-5 Symmetry
Every transformation has a pre-image and an image.
• Pre-image is the original figure in the transformation (the “before”). Its points are labeled as usual.
• Image is the shape that results from the transformation (the “after”). The points are labeled with the same letters but with a ' (prime) symbol after each letter.
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Holt Geometry
12-5 Symmetry
symmetryline symmetryline of symmetryrotational symmetry
Vocabulary
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Holt Geometry
12-5 Symmetry
A figure has symmetry if there is a transformation of the figure such that the image coincides with the preimage.
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Holt Geometry
12-5 Symmetry
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Holt Geometry
12-5 Symmetry
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: 4
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Holt Geometry
12-5 Symmetry
Example 1: Identifying symmetry
Tell whether the figure has line symmetry. If so, draw all lines of symmetry.
yes; one line of symmetry
Tell whether the figure has rotational symmetry. If so, give the angle of symmetry.
no;
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Holt Geometry
12-5 Symmetry
Example 1: Identifying symmetry
Tell whether the figure has line symmetry. If so, draw all lines of symmetry.
No;
Tell whether the figure has rotational symmetry. If so, give the angle of symmetry.
Yes; 180°
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Holt Geometry
12-5 Symmetry
Example 1: Identifying symmetry
Tell whether the figure has line symmetry. If so, draw all lines of symmetry.
No;
Tell whether the figure has rotational symmetry. If so, give the angle of symmetry.
Yes; 180°
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Holt Geometry
12-5 Symmetry
Example 1: Identifying symmetry
Tell whether the figure has line symmetry. If so, draw all lines of symmetry.
yes; one line of symmetry
Tell whether the figure has rotational symmetry. If so, give the angle of symmetry.
no;
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Holt Geometry
12-5 Symmetry
Symmetry and Coordinate Plane
• To write an equation you need to know its
slope and y-intercept.
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Holt Geometry
12-5 SymmetryWriting Equations for Lines of Symmetry
Remember equations for vertical lines:
x = 2 is vertical line crossing x-axis at 2x = –4 is vertical line crossing x-axis at –4
x=2x =–4
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Holt Geometry
12-5 SymmetryWriting Equations for Lines of Symmetry
Remember equations for horizontal lines:
y = 2 is horizontal line crossing y-axis at 2y = –4 is horizontal line crossing y-axis at 4
y=2
y =–4
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Holt Geometry
12-5 SymmetryWriting Equations for Lines of Symmetry
Writing equations in form of y=mx + b
m is the slope (rise/run)
3
2
m =3 2
b = 3
Equation of line: y = 3x + 3 2
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Holt Geometry
12-5 Symmetry
13.
“Symmetry Worksheet” Practice Problems
Equation for line of symmetry_________________
Crosses
XX – axis at 3
x = 3
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Holt Geometry
12-5 Symmetry
“Symmetry Worksheet” Practice Problems
Equation: ___________________
14.
Write the equation of the line of symmetry
Crosses
y – axis at -5
y = -5
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Holt Geometry
12-5 Symmetry
•Identify tessellations •Know if figures will tessellate
Tessellation Objectives
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Holt Geometry
12-5 Symmetry
A tessellation, or tiling, is a repeating pattern that completely covers a plane with no gaps or overlaps. The measures of the angles that meet at each vertex must add up to 360°.
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Holt Geometry
12-5 Symmetry
What can Tessellate?• Try to tessellate these shapes
A. B.
NoYes
YesNo
C. D.
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Holt Geometry
12-5 Symmetry
A pattern has tessellation symmetry if it can be translated along a vector so that the image coincides with the preimage. A frieze pattern is a pattern that has translation symmetry along a line.
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Holt Geometry
12-5 Symmetry
A glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector.
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Holt Geometry
12-5 Symmetry
Both of the frieze patterns shown below have translation symmetry. The pattern on the right also has glide reflection symmetry. A pattern with glide reflection symmetry coincides with its image after a glide reflection.
Practice seeing the patterns
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Holt Geometry
12-5 Symmetry
A regular tessellation is formed by congruent regular polygons. A semiregular tessellation is formed by two or more different regular polygons, with the same number of each polygon occurring in the same order at every vertex.
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Holt Geometry
12-5 Symmetry
Regulartessellation
Semiregulartessellation
Every vertex has two squares and three triangles in this order: square, triangle, square, triangle, triangle.
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Holt Geometry
12-5 Symmetry
Check It Out! Example 1
Identify the symmetry in each frieze pattern.
a. b.
translation symmetrytranslation symmetry and glide reflection symmetry
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Holt Geometry
12-5 Symmetry
Check It Out! Example 3
Classify each tessellation as regular, semiregular, or neither.
Only hexagons are used. The tessellation is regular.
It is neither regular nor semiregular.
Two hexagons meet two triangles at each vertex. It is semiregular.