Holt Geometry

12-5 Symmetry

COMPOSITIONS OF TRANSFORMATIONS

• I can perform a composition of two or more transformations.

Holt Geometry

12-5 Symmetry

How are transformations used?

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******

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)

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)

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)

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)

Holt Geometry

12-5 Symmetry

Brain Stretch

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

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.

Holt Geometry

12-5 Symmetry

symmetryline symmetryline of symmetryrotational symmetry

Vocabulary

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.

Holt Geometry

12-5 Symmetry

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

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;

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°

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°

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;

Holt Geometry

12-5 Symmetry

Symmetry and Coordinate Plane

• To write an equation you need to know its

slope and y-intercept.

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

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

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

Holt Geometry

12-5 Symmetry

13.

“Symmetry Worksheet” Practice Problems

Equation for line of symmetry_________________

Crosses

XX – axis at 3

x = 3

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

Holt Geometry

12-5 Symmetry

•Identify tessellations •Know if figures will tessellate

Tessellation Objectives

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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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12-5 Symmetry

What can Tessellate?• Try to tessellate these shapes

A. B.

NoYes

YesNo

C. D.

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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.

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.

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

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.

Holt Geometry

12-5 Symmetry

Regulartessellation

Semiregulartessellation

Every vertex has two squares and three triangles in this order: square, triangle, square, triangle, triangle.

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

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.