COMPOSITIONS OF TRANSFORMATIONS I can perform a composition of two or more transformations.

How are transformations used?

Compositions of Transformations One 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******

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’(2, -6) B’(7, -3) C’(3, 2) A’’(6, 2) B’’(3, 7) C’’(-2, 3) A’ B’ C’

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)

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’(0, 10) B’(5, 7) C’(1, 2) A’’(0, -10) B’’(5, -7) C’’(1, -2) A’’ B’’ C’’

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)

Brain Stretch

Symmetry 12-5 I 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

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.

Vocabulary symmetry line symmetry line of symmetry rotational symmetry

A figure has symmetry if there is a transformation of the figure such that the image coincides with the preimage.

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

Example 1: Identifying symmetry Tell whether the figure has line symmetry. If so, draw all lines of symmetry. Tell whether the figure has rotational symmetry. If so, give the angle of symmetry. yes; one line of symmetry no;

Example 1: Identifying symmetry Tell whether the figure has line symmetry. If so, draw all lines of symmetry. Tell whether the figure has rotational symmetry. If so, give the angle of symmetry. No; Yes; 180°

Example 1: Identifying symmetry Tell whether the figure has line symmetry. If so, draw all lines of symmetry. Tell whether the figure has rotational symmetry. If so, give the angle of symmetry. No; Yes; 180°

Example 1: Identifying symmetry Tell whether the figure has line symmetry. If so, draw all lines of symmetry. Tell whether the figure has rotational symmetry. If so, give the angle of symmetry. yes; one line of symmetry no;

Symmetry and Coordinate Plane To write an equation you need to know its slope and y-intercept.

Writing Equations for Lines of Symmetry Remember equations for vertical lines: x = 2 is vertical line crossing x-axis at 2 x = –4 is vertical line crossing x-axis at –4 x=2 x =–4

Writing Equations for Lines of Symmetry Remember equations for horizontal lines: y = 2 is horizontal line crossing y-axis at 2 y = –4 is horizontal line crossing y-axis at 4 y=2 y =–4

Writing Equations for Lines of Symmetry Writing equations in form of y=mx + b m is the slope (rise/run) 2 m =3 2 3 b = 3 Equation of line: y = 3x + 3 2

“Symmetry Worksheet” Practice Problems 13. Crosses X – axis at 3 x = 3 Equation for line of symmetry_________________

“Symmetry Worksheet” Practice Problems Write the equation of the line of symmetry 14. Crosses y – axis at -5 Equation: ___________________ y = -5

Tessellation Objectives Identify tessellations Know if figures will tessellate

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°.

What can Tessellate? Try to tessellate these shapes A. B. Yes No C. D.

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.

A glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector.

Both of the frieze patterns shown below have translation 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

A regular tessellation is formed by congruent regular polygons 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.

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

Check It Out! Example 1 Identify the symmetry in each frieze pattern. a. b. translation symmetry and glide reflection symmetry translation 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.