Properties of Transformations

Translate Figures and Use Vectors Translation: moves every point of a figure the same distance in the same direction Image: the new figure formed by the translation Preimage: another name for the original figure Isometry: a transformation that preserves length and angle measurements

Graph quadrilateral ABCD with vertices A (–1, 2), B (–1, 5), C (4, 6), and D (4, 2). Find the image of each vertex after the translation (x, y) → (x + 3, y – 1). Then graph the image using prime notation. SOLUTION First, draw ABCD. Find the translation of each vertex by adding 3 to its x -coordinate and subtracting 1 from its y -coordinate. Then graph the image.

Graph quadrilateral ABCD with vertices A (–1, 2), B (–1, 5), C (4, 6), and D (4, 2). Find the image of each vertex after the translation (x, y) → (x + 3, y – 1). Then graph the image using prime notation. (x, y) → (x + 3, y – 1) A (–1, 2) → A′ (2, 1 ) B (–1, 5) → B′ (2, 4) C (4, 6) → C′ (7, 5) D (4, 2) → D′ (7, 1)

Draw RST with vertices R (2, 2), S (5, 2), and T (3, 5). Find the image of each vertex after the translation (x, y) → (x + 1, y + 2). Graph the image using prime notation. SOLUTION First, draw. Find the translation of each vertex by adding 1 to its x -coordinate and adding 2 to its y -coordinate. Then graph the image. RST

Graph triangle RST with vertices R (2, 2), S (5, 2), T (3, 5). Find the image of each vertex after the translation (x, y) → (x + 1, y + 2). Then graph the image using prime notation. (x, y) → (x + 1, y + 2) R (2, 2) → R′ (3, 4) S (5, 2) → S′ (6, 4) T (3, 5) → T′ (4, 7) y RS T R ′ S ′ T ′ x

The image of (x, y) → (x + 4, y – 7) is P′Q′ with endpoints P′ (–3, 4) and Q′ (2, 1). Find the coordinates of the endpoints of the preimage. SOLUTION P' ( 3, 4) → P (– 7, 11) - Q' (2, 1) → Q (– 2, 8)

Write a rule for the translation of ABC to A ′ B ′ C ′. Then verify that the transformation is an isometry.

To go from A to A ′, move 4 units left and 1 unit up. So, a rule for the translation is (x, y) →(x – 4, y + 1). SOLUTION

Vectors: a quantity that has both direction and magnitude and is represented in the coordinate plane by an arrow drawn from one point to another. Initial point: the starting point of the vector Terminal point: the ending point of the vector

Component form : combines the horizontal and vertical components Horizontal component goes left to right. Vertical component goes up and down.

Name the vector and write its component form. SOLUTION The vector is BC. From initial point B to terminal point C, you move 9 units right and 2 units down. So, the component form is 9, –2.

Name the vector and write its component form. The vector is ST. From initial point S to terminal point T, you move 8 units left and 0 units vertically. The component form is –8, 0. SOLUTION

Name the vector and write its component form. The vector is TX. From initial point T to terminal point S, you move 0 units horizontally and 3 units up. The component form is 0, 3. SOLUTION

The vertices of ∆ LMN are L(2, 2), M(5, 3), and N(9, 1). Translate ∆ LMN using the vector –2, 6. SOLUTION Find the translation of each vertex by subtracting 2 from its x -coordinate and adding 6 to its y -coordinate. (x, y) → (x – 2, y + 6) L(2, 2) → L ′ (0, 8) M(5, 3) → M ′ (3, 9) N(9, 1) → N ′ (7, 7)

Lesson – 9.3 Perform Reflections Reflection: a transformation that uses a line like a mirror to reflect an image Line of reflection: a line that acts like a mirror to reflect an image.

SOLUTION The vertices of ABC are A(1, 3), B(5, 2), and C(2, 1). Graph the reflection of ABC described. In the line m : y = 1 Point A is 2 units above m, so A ′ is 2 units below m at (1, –1). Also, B ′ is 1 unit below m at (5, 0). Because point C is on line m, you know that C = C ′.

Graph a reflection of ABC from Example 1 in the given line. y = 4 ANSWER

Graph a reflection of ABC from Example 1 in the given line. x = -3 ANSWER

Coordinate Rules for Reflections If (a,b) is reflected in the x-axis, its image is the point (a,-b). If (a,b) is reflected in the y-axis, its image is the point (-a,b). If (a,b) is reflected in the line y = x, its image is the point (b,a). If (a,b) is reflected in the line y = - x, its image is the point (-b,-a).

The endpoints of FG are F(–1, 2) and G(1, 2). Reflect the segment in the line y = x. Graph the segment and its image.

The slope of GG ′ will also be –1. From G, move 0.5 units right and 0.5 units down to y = x. Then move 0.5 units right and 0.5 units down to locate G ′ (2, 1). SOLUTION The slope of y = x is 1. The segment from F to its image, FF ′, is perpendicular to the line of reflection y = x, so the slope of FF ′ will be –1 (because 1(–1) = –1 ). From F, move 1.5 units right and 1.5 units down to y = x. From that point, move 1.5 units right and 1.5 units down to locate F ′ (2,–1).

SOLUTION Use the coordinate rule for reflecting in y = –x. (a, b) (–b, –a) G(1, 2) G ′ (–2, –1) F(–1, 2)F ′ (–2, 1) Reflect FG from Example 2 in the line y = –x. Graph FG and its image.

Graph ABC with vertices A(1, 3), B(4, 4), and C(3, 1). Reflect ABC in the lines y = –x and y = x. Graph each image. SOLUTION

Homework: p {1-12}

Lesson 9.6 -Identify Symmetry Line Symmetry: occurs in a figure in a plane if the figure can be mapped onto itself by a reflection in a line. Line of Symmetry: The line of reflection that maps a figure onto itself Rotational symmetry: a figure that can be mapped onto itself by a rotation of or less about the center of the figure

SOLUTION How many lines of symmetry does the hexagon have? a. b.c. a. Two lines of symmetry. b. Six lines of symmetry. c. One line of symmetry. SOLUTION

How many lines of symmetry does the object appear to have? ANSWER 8 5 1

Draw a hexagon with no lines of symmetry. ANSWER

Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. Parallelogram SOLUTION a. The parallelogram has rotational symmetry. The center is the intersection of the diagonals. A 180° rotation about the center maps the parallelogram onto itself.

b. The regular octagon has rotational symmetry.The center is the intersection of the diagonals. Rotations of 45, 90, 135, or 180° about the center all map the octagon onto itself. o o o b. Regular octagon SOLUTION

c. The trapezoid does not have rotational symmetry because no rotation of 180° or less maps the trapezoid onto itself. c. Trapezoid SOLUTION

Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. Rhombus b. Octagon c. Right triangle SOLUTION yes; 180° about the center SOLUTION yes; 90° or 180° about the center SOLUTION no

Homework: p621: {3-5,6,13,14,17,18}

Section 9.4 – Perform Rotations Rotation : a transformation in which a figure is turned about a fixed point Center of Rotation : the fixed point in which a figure is turned about Angle of Rotation : formed from rays drawn from the center of rotation to a point and its image.

SOLUTION STEP 1 Draw a segment from A to P. Draw a 120 rotation of ABC about P. o

STEP 2 STEP 3 Draw A ′ so that PA ′ = PA. Draw a ray to form a 120 angle with PA. o

STEP 4 Repeat Steps 1– 3 for each vertex. Draw A ′ B ′ C ′.

Coordinate Rules for Rotations about the origin When a point (a,b) is rotated counterclockwise about the origin, the following are true: 1. For a rotation of 90, (a,b) becomes (-b,a). 2. For a rotation of 180, (a,b) becomes (-a,-b). 3. For a rotation of 270, (a,b) becomes (b,-a).

Graph the image R ′ S ′ T ′ U ′. SOLUTION (a, b) (b, –a) R(3, 1) R ′ (1, –3) T(5, –3) T ′ (–3, –5) U(2, –1) U ′ (–1, –2) S(5, 1)S ′ (1, –5) Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, –3), and U(2, –1). Then rotate the quadrilateral 270 about the origin. o Graph RSTU. Use the coordinate rule for a 270 rotation to find the images of the vertices. o

Trace DEF and P. Then draw a 50° rotation of DEF about P. ANSWER

Graph JKL with vertices J(3, 0), K(4, 3), and L(6, 0). Rotate the triangle 90° about the origin. ANSWER

SOLUTION By Theorem 9.3, the rotation is an isometry, so corresponding side lengths are equal. Then 2x = 6, so x = 3. Now set up an equation to solve for y. Corresponding lengths in an isometry are equal. Substitute 3 for x. Solve for y. 3x + 15y5y = 5y5y = 3(3) + 1 = y2 The correct answer is B.

Find the value of r in the rotation of the triangle. The correct answer is B. ANSWER

HOMEWORK: p {1,3-14,20,21,23

TESSELLATIONS HOW TO DO IT YOURSELF! SLICE METHOD

Slice Method - Example 1 First a very simple one to show the principle. 1. Start with a shape that is known to tessellate - in this case, a square...

Slice Method - Example 1 2. Mark out a shape to be cut out on one side...

Slice Method - Example 1 3. Cut out the slice and place it on the opposite side...

Slice Method - Example 1 4. This shape will tessellate like this...

Slice Method - Example 1 5. And it's possible to make a simple picture with it. It will still tessellate if the piece is reversed or stuck on the side. It will even do so if a piece is taken from the side and the top together... there are several combinations but some will not work.

Slice Method - Example 2 Begin with a shape that will tessellate. This is a hexagon with opposite sides equal. Such a shape is always OK...

Slice Method - Example 2 2. Draw a random squiggly line along one side making sure it ends at the corners...

Slice Method - Example 2 3. Cut out the slice and paste it along the outside of the opposite side...

Slice Method - Example 2 4. Draw a random line on another side...

Slice Method - Example 2 5. Cut the slice out and paste it on the opposite side as before...

Slice Method - Example 2 6. Repeat for the remaining side...

Slice Method - Example 2 7. Look at the resulting shape - any ideas yet?

Slice Method - Example 2 8. To give a clean shape, remove all the bits outside the lines and fill in the corresponding parts inside. What is it?

Slice Method - Example 2 9. This is what I saw! As you see, the shape was arrived at quite by accident but I acknowledge the Disney copyright of my artistic filling in. Will it tessellate?

Slice Method - Example Yes, of course! Can we improve it?

Slice Method - Example Add a road, wall and pavement and we have a parade!

YOUR TURN! GOOD LUCK!!!

Section 9.5: Apply Compositions of ransformations Glide Reflection : A transformation in which every point P is mapped to a point P” by the following steps. 1. a translation maps P to P’ 2. a reflection in a line k parallel to the direction of the translation maps P’ to P”

SOLUTION The vertices of ABC are A(3, 2), B(6, 3), and C(7, 1). Find the image of ABC after the glide reflection. Translation : (x, y) → Reflection: in the x -axis (x –12, y) Begin by graphing ABC. Then graph A ′ B ′ C ′ after a translation 12 units left. Finally, graph A ′′ B ′′ C ′′ after a reflection in the x -axis.

Suppose ABC in Example 1 is translated 4 units down, then reflected in the y -axis. What are the coordinates of the vertices of the image? SOLUTION A(3, 2) → A ′ (3, – 2) B(6, 3) → B ′ (6, – 1) C(7, 1) → C ′ (7, – 3) ( x, y ) (x, y – 4 ) (x, y) → (–a, b) Reflection: in the y- axis → A " (–3, – 2) → B " (–6, – 1) → C " (–7, – 3) Translation: ( x, y ) (x, y – 4 ) → (–a, b)

In Example 1, describe a glide reflection from A ′′ B ′′ C ′′ to ABC. SOLUTION Reflection: in the x -axis Translation : (x, y) → ( x +12, y) Begin by graphing A ′ B ′ C ′. Then graph ABC after a translation 12 units right. Finally, graph ABC after a reflection in the x -axis.

Composition of Transformations: the result of two or more transformations that are combined to form a single transformation

SOLUTION Reflection: in the y -axis Rotation: 90° about the origin STEP 1 Graph RS Reflect RS in the y -axis. R ′ S ′ has endpoints R ′ (–1, –3) and S ′ (–2, –6). STEP 2 The endpoints of RS are R(1, –3) and S(2, –6). Graph the image of RS after the composition.

STEP 3 Rotate R ′ S ′ 90 about the origin. R ′′ S ′′ has endpoints R ′′ (3, –1) and S ′′ (6, –2). o

Graph RS from Example 2. Do the rotation first, followed by the reflection. Does the order of the transformations matter? Explain. SOLUTION Yes; the resulting segment R ′′ S ′′ is not the same.

HOMEWORK p {1, 3-14, 20, 21, 23} p {1-4, 7, 9, 11, 13, 14}

Lesson 9.6 -Identify Symmetry Line Symmetry occurs in a figure in a plane if the figure can be mapped onto itself by a reflection in a line. Line of Symmetry: The line of reflection that maps a figure onto itself Rotational symmetry: a figure that can be mapped onto itself by a rotation of or less about the center of the figure

SOLUTION How many lines of symmetry does the hexagon have? a. b.c. a. Two lines of symmetry. b. Six lines of symmetry. c. One line of symmetry. SOLUTION

How many lines of symmetry does the object appear to have? ANSWER 8 5 1

Draw a hexagon with no lines of symmetry. ANSWER

Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. Parallelogram SOLUTION a. The parallelogram has rotational symmetry. The center is the intersection of the diagonals. A 180° rotation about the center maps the parallelogram onto itself.

b. The regular octagon has rotational symmetry.The center is the intersection of the diagonals. Rotations of 45, 90, 135, or 180° about the center all map the octagon onto itself. o o o b. Regular octagon SOLUTION

c. The trapezoid does not have rotational symmetry because no rotation of 180° or less maps the trapezoid onto itself. c. Trapezoid SOLUTION

Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. Rhombus b. Octagon c. Right triangle SOLUTION yes; 180° about the center SOLUTION yes; 90° or 180° about the center SOLUTION no

Homework: p621: {3-5,6,13,14,17,18}

Lesson 9.7 Identify and Perform Dilations Dilation: a transformation that stretches or shrinks a figure to create a similar figure. Center of Dilation: the fixed point Scale Factor of a Dilation: the ratio of a side length of the image to the corresponding side length of the original figure.

Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. a. SOLUTION a. Because =, the scale factor is k =. The image P’ is an enlargement. CP’ CP

Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. b. SOLUTION Because = the scale factor is k =. The image P’ is a reduction. CP’ CP 18 30’ 3 5 b.

Draw and label DEFG. Then construct a dilation of DEFG with point D as the center of dilation and a scale factor of 2. SOLUTION STEP 1 Draw DEFG. Draw rays from D through vertices E, F, and G.

STEP 2 Open the compass to the length of DE. Locate E’ on DE so DE’ = 2(DE). Locate F’ and G’ the same way.

STEP 3 Add a second label D’ to point D. Draw the sides of D’E’F’G’.

In a dilation, CP’ = 3 and CP = 12. Tell whether the dilation is a reduction or an enlargement and find its scale factor. Because =, the scale factor is k =. The image P’ is a reduction. CP’ CP ANSWER

Draw and label RST. Then construct a dilation of RST with R as the center of dilation and a scale factor of 3. ANSWER

Homework: p {3-5,7,11,13,21,22,23 }