9.5 & 9.6 – Compositions of Transformations & Symmetry

Glide Reflection: A transformation with a translation and then a reflection in a line parallel to the direction of the translation k Q P

Composition of Transformations: Any two transformations combined to form a single transformation

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a ___________. If ____ is the image of P after the two reflections, then: translation

If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a _________. If ____ is the image of P after the two reflections, then: rotation The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by k and m

A 1. Translation: (x, y) → (x + 2, y) Reflection: in the x-axis Graph the image of A(1, –2) after the described glide reflection. 1. Translation: (x, y) → (x + 2, y) Reflection: in the x-axis A

A 2. Translation: (x, y) → (x – 1, y + 3) Reflection: in x = 2 Graph the image of A(1, –2) after the described glide reflection. 2. Translation: (x, y) → (x – 1, y + 3) Reflection: in x = 2 A

(x, y) → (–y, x) B (0, 0) (0, 0) C (–2, 4) A (–4, –2) (2, 2) (–2, 2) The vertices of ABC are A(3, 1), B(1, 5) and C(5, 3). Graph the image of ABC after a composition of the transformations in the order they are listed. Translation: (x, y) → (x – 3, y – 1) Rotation: 90° (x, y) → (–y, x) B (0, 0) (0, 0) C (–2, 4) A (–4, –2) (2, 2) (–2, 2)

(x, y) → (y, x) B (3, 1) (1, 3) C (1, 5) A (5, 1) (5, 3) (3, 5) The vertices of ABC are A(3, 1), B(1, 5) and C(5, 3). Graph the image of ABC after a composition of the transformations in the order they are listed. Reflection: y = x Reflection: y = 1 (x, y) → (y, x) B (3, 1) (1, 3) C (1, 5) A (5, 1) (5, 3) (3, 5)

Describe the composition of transformations. Reflection: x-axis Translation: (x, y) → (x + 6, y + 2)

Describe the composition of transformations. Rotation: 90° Reflection: x-axis

In the diagram, is reflected in line r, and is reflected in line s. 7. A translation maps onto which segment?

In the diagram, is reflected in line r, and is reflected in line s. 8. Which lines are perpendicular to r and s

In the diagram, is reflected in line r, and is reflected in line s. 9. Name a segment parallel to

In the diagram, is reflected in line r, and is reflected in line s. 10. If the distance between r and s is 2.4 inches, what is the length of 2.4  2 4.8 inches

In the diagram, is reflected in line r, and is reflected in line s. 11. Is the distance from to r the same as the distance from C to r? Yes, Def. of Reflection

12. Find the angle of rotation that maps T onto 75  2 150°

13. Find the angle of rotation that maps A onto 99  2 198°

Line of Symmetry: The figure can be mapped onto itself by a reflection in the line Rotational Symmetry: The figure can be mapped onto itself by a rotation of 180° or less around a point

How many lines of symmetry does the triangle have? one

How many lines of symmetry does the triangle have? none

How many lines of symmetry does the triangle have? Three

Determine whether the figure has rotational symmetry Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. Yes, 90° and 180° 360 4 = 90°

Determine whether the figure has rotational symmetry Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. Yes, 360 6 = 60° 60°, 120°, 180°

Determine whether the figure has rotational symmetry Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. No

Determine whether the figure has rotational symmetry Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. Yes, 360 8 = 45° 45°, 90°, 135°, 180°

Identify the line symmetry and rotational symmetry of the figure shown. 4 Rotational: 90°, 180° 360 4 = 90°

Identify the line symmetry and rotational symmetry of the figure shown. 1 Rotational: none

Identify the line symmetry and rotational symmetry of the figure shown. 2 Rotational: 180°,