A transformation is a function, or mapping, that results in a change in the position, shape, or size of the figure.

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1 Translations Geometry Unit 9: Lesson 1 Name A transformation is a function, or mapping, that results in a change in the position, shape, or size of the figure. Some basic transformations include translations, reflections, and rotations. slide flip turn The original figure is the preimage. The resulting figure is the image. Some transformations (translations, reflections, and rotations) preserve distance and angle measures. To preserve distance means the distance between any two points of the image is the same as the distance between the corresponding points of the preimage. (The length of the sides has not changed.) To preserve angles means that the angles of the image have the same angle measure as the corresponding angles of the preimage. (The measures of the angles has not changed.) A transformation that preserves distance and angle measures is called a rigid motion or an isometry. A transformation maps every point of a figure onto its image and may be described with arrow notation,. Prime notation,, is sometimes used to identify image points. List corresponding points of the preimage and image in the same order, just as for congruent or similar figures. A translation is a transformation that maps all points of a figure the same distance in the same direction.

Notation: (x, y) (x + 4, y 2) OR T4, -2(ABCD) = A B C D OR 4, 2 Each point of the preimage moves 4 units right and 2 units down.

2 Since a translation maps all points the same distance in the same direction, AA = BB = CC. (Also AA BB CC ) Translations preserve distance. AB = A B BC = B C AC = A C Translations preserve angle measurement. m A m A' m B m B' m C m C' A translation is a rigid motion (isometry). Notation: (x, y) (x + 4, y 2) OR T4, -2(ABCD) = A B C D OR 4, 2 Each point of the preimage moves 4 units right and 2 units down. Another way to describe a translation is with a vector. A vector is a quantity that has a magnitude (length) and direction. The direction is indicated by an arrow pointing from the tail (initial point) to the head (terminal point). AB

3 The vector that defines the following translation is drawn. The vector may be described as <4, -9>. This is called component form. A composition of transformations is a combination of two or more transformations. In a composition, you perform each transformation on the image of the preceding transformation. In general, the composition of two translations is another translation. 1. Does the transformation appear to be a rigid motion? Explain. a) b) 2. a) A is a transformation that slides every point of a preimage the same distance in the same direction to its image. b) A translation is a function that takes every point of a preimage and moves it along a to its image. 3. Which of the following properties are invariant under a translation? a) distance (or betweeness) b) angle measure c) orientation d) perimeter e) area f) collinearity (points collinear on preimage are collinear on the image) g) parallelism (parallel lines/segments on preimage are parallel on the image) h) perpendicularity (perpendicular lines/segments on preimage are perpendicular on the image) i) Is a translation a direct or opposite isometry?

b) Write the vector in component form. c) Write a rule that describes the translation that maps PQRS onto P Q R S? 6.

4 4. Given PQR with coordinates P(2, 1), Q(3, 3), and R(-1, 3). a) What are the vertices of T-2, -5( PQR )? Graph PQR and its image. b) Draw PP ', QQ ', and RR '. What relationships exist among these three segments? 5. a) Draw the vector that defines the translation. b) Write the vector in component form. c) Write a rule that describes the translation that maps PQRS onto P Q R S? 6. The diagram shows two moves of the black bishop in a chess game. a) Write the translation rules to represent each move. b) Write a single translation equivalent to the composition of the bishop s two moves. B B

Reflections Geometry Unit 9: Lesson 2 Name When you reflect a figure across a line, each point of the figure maps to another point the same distance from the line, but on the other side.

5 Reflections Geometry Unit 9: Lesson 2 Name When you reflect a figure across a line, each point of the figure maps to another point the same distance from the line, but on the other side. The orientation of the figure reverses. For a reflection across line m, the line of reflection: If a point A is on line m, then the image of A is itself (A = A). If a point B is not on line m, then m is the perpendicular bisector of BB Properties of Reflections: Reflections preserve distance. AB = A B. Reflections preserve angle measurement. m ABC m A' B' C' A reflection is a rigid motion (isometry). A reflection reverses orientation. Notation: rm (C ) = C. A reflection across m that takes C to C. ry = x reflection across the line y = x rx-axis reflection across the x-axis rx-axis (x, y) = (x, -y) ry-axis (x, y) = (-x, y) ry = x (x, y) = (y, x) ry = -x (x, y) = (-y, -x) rorigin (x, y) = (-x, -y) Best way: 1. Plot points 2. Graph the line of reflection (dotted) 3. Count from each point to the line of reflection & count the same number away from the line of reflection to get the prime point. MUST KNOW Remember to graph y = #... Go to the # on the y-axis and put a dot. Then graph the line HORIZONTAL Remember to graph x = #... Go to the # on the x-axis and put a dot. Then graph the line VERTICAL

6 1. The line of reflection is the of the line segment connecting a point to its image. 2. Which of the following properties are invariant under a line reflection? a) distance (or betweeness) b) angle measure c) orientation d) perimeter e) area f) collinearity g) parallelism h) perpendicularity i) Is a line reflection a direct or opposite isometry? 3. a) Graph P(3, 4). b) Graph y = 1. c) What are the coordinates of ry = 1(P)? Graph this point. d) Draw PP '. e) Find the midpoint of PP '. f) Find the slope of PP '. g) Find the slope of y = 1. h) PP ' and y = 1 are f) The line of reflection, y = 1, is the of PP '.

7 4. Graph points A(-3, 4), B(0, 1), and C(4, 2). Graph and label ry-axis( ABC ). 5. Graph points A(-3, 4), B(0, 1), and C(4, 2). Graph and label rx-axis( ABC ). 6. Each triangle in the diagram is a reflection of another triangle across one of the given lines. a) How can you describe Triangle 2 by using a reflection rule? b) How can you describe Triangle 1 by using a reflection rule? 7. In the diagram, rt (G) = G, rt (H) = J, and rt (D) = D. Use the properties of reflections to describe how you know that GHJ is an isosceles triangle.

8 Rotations Geometry Unit 9: Lesson 3 Name: A rotation is a transformation that turns a figure around a point, called the center of rotation. The number of degrees a figure rotates is considered the angle of rotation. The counterclockwise direction is a positive angle of rotation. C B A C B Angle of Rotation B A B x A C Center of Rotation A C The distance between the center of rotation and A is equal to the distance between the center of rotation and A. B C A B Notation: RC, 90 (P ) = P Rotate P 90 about center of rotation C. Properties of Rotations Rotations preserve distance. AB = A B. Rotations preserve angle measurement. m ABC m A' B' C' A rotation about a point is a rigid motion (isometry). A C Rotations in the coordinate plane.

RO, 90 (x, y) (-y, x) RO, 180 (x, y) (-x, -y) RO, 270 (x, y) (y, -x) RO, -90 = RO, 270 Rotation about the origin is equivalent to a reflection through the origin.

Find the coordinates of the points now these are the prime points 4. Turn back to the original position and plot the new points (Prime points) 1.

rotation is to the distance between the corresponding point on the image and the center of rotation.

9 RO, 90 (x, y) (-y, x) RO, 180 (x, y) (-x, -y) RO, 270 (x, y) (y, -x) RO, -90 = RO, 270 Rotation about the origin is equivalent to a reflection through the origin. Best way to perform a rotation centered at the origin: 1. Plot the pre-image 2. Turn the paper LEFT 90 o --- Once 180 o --- Twice 270 o --- Three Times 3. Find the coordinates of the points now these are the prime points 4. Turn back to the original position and plot the new points (Prime points) 1. a) A rotation turns a preimage a given number of with respect to a b) A rotation of a positive degree measure turns a figure c) The distance between a point on the preimage and the center of rotation is to the distance between the corresponding point on the image and the center of rotation. d) Under a rotation, the angle formed by a point, the center of rotation, and the corresponding image of the point is equal to the e) A rotation of 180 is equivalent to a reflection. 2. Which of the following properties are invariant under a rotation? a) distance (or betweeness) b) angle measure c) orientation d) perimeter e) area f) collinearity g) parallelism h) perpendicularity i) Is a rotation a direct or opposite isometry?

10 3. a) PQRS has vertices P(1,1), Q(3, 3), R(4, 1), and S(3, 0). Graph PQRS and RO, 90 (PQRS). b) Connect P to the origin, O, (center of rotation) and the origin to P'. What is m POP'? c) Find m QOQ', m ROR', m SOS'. d) Complete the following: PO OP' QO OQ' RO OR', SO OS' 4. a) PQRS has vertices P(1,1), Q(3, 3), R(4, 1), and S(3, 0). Graph PQRS and RO, 180 (PQRS). b) Find m POP'. c) PO OP' d) A rotation of 180 is equivalent to a point reflection through which point?

11 5. PQRS has vertices P(1,1), Q(3, 3), R(4, 1), and S(3, 0). Graph PQRS and RO, 270 (PQRS). 6. WXYZ is a parallelogram, and T is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of opposite sides of the parallelogram are equal? W Z T X Y

12 Geometry Notes Rotations Unit 9: Lesson 4 Name: 1. What is the image of RP, 90 ( ABC )? STEPS: 1. Connect A to P (center of rotation) 2. Construct the angle you need 3. Make an arc through the angle 4. Take that measurement & put the compass point on P and make a circle 5. Measure the opening of the angle 6. Put the point where the line you drew passes through the circle make an arc on the circle. (Down for counterclockwise) 7. Draw a line from the center of rotation through where your arc passed through the circle. 8. Measure from the center of rotation to pointa 9. Put this measurement on the line that you just drew. This is A 10. Repeat for the other points P B A C

13 2. What is the image of RP, 60 ( ABC )? P B A C

14 Compositions of Isometries Geometry Unit 9: Lesson 5 Name Isometry Rigid motion, preserves distance and angle measure. A composition of transformations is a combination of two or more transformations performed one after another. Compositions can be expressed in the following ways. (T 2,0 r y=x )(A) = A T 2,0 (r y=x (A)) = A do this second do this first do this second do this first The composition of reflections across two parallel lines is a translation. ( r r )( ABC) A'' B'' C'' m l The distance of the translation is twice the distance between the lines. The composition of reflections across two intersecting lines is a rotation. ( r r )( ABC) A'' B'' C'' m l The center of rotation is Q and the angle of rotation is twice the acute angle measure.

15 Glide Reflection Composition of a translation and reflection, such that the translation must be parallel to the line of reflection. The order in which a glide reflection is performed does not matter to obtain the resulting image, however this is not true of other compositions. 1. Graph PNB with vertices P(2, 2), N(3, -1), and B(-1, -2) and its image after the transformation (ry = 3(T2, 0( PNB ))).

16 2. a) Graph TEX with vertices T(-5, 2), E(-1, 3), and X(-2, 1). What is the image of TEX for (rx = 0 T0, -5)( TEX )? b) What type of a composition of isometries is this? Explain. 3. a) What is (rm rl)(j)? b) The composition of a reflection across two parallel lines is equivalent to what single transformation? c) What is the distance of the resulting translation?

transformation? c) What are center and angle of rotation for the resulting rotation? 5.

17 4. a) What is (rb ra)(j)? b) The composition of a reflection across two intersecting lines is equivalent to what transformation? c) What are center and angle of rotation for the resulting rotation? 5. Describe the composition of isometries that maps the black figure to the gray figure. 6. Describe the composition of isometries that maps the black figure to the gray figure.

18 7. Identify each mapping as a translation, reflection, rotation, or glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation followed by a reflection. a) ABC EDC b) EDC PQM c) MNJ EDC d) HIF HGF e) PQM JLM

19 Congruence Transformations Geometry Unit 9: Lesson 6 Name Rigid Motion - Compositions of rigid motions can be used to understand congruence. Two figures are congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto the other. Compositions of rigid motions take figures to congruent figures. Therefore, they are called congruence transformations. 1. Graph PNB with vertices P(2, 2), N(3, -1), and B(-1, -2) and its image after the transformation (R 0,90 ry =3( PNB )). Is this a congruence transformation? Explain.

20 1. a) If two figures are congruent, then there is a sequence of that map one figure onto the other. b) If there is a sequence of rigid motions that map one figure onto another, then the two figures are 2. The composition (rn RP, 90)(LMNO) = GHJK. a) Which angle pairs have equal measures? b) Which sides have equal lengths? 3. a) Describe the sequence of rigid motions that maps EDF onto MLN. b) Describe the sequence of rigid motions that maps XWZY onto BAJC. c) Describe the sequence of rigid motions that maps PQ onto HG.

21 4. In the diagram, JQV EWT. What is a congruence transformation that maps JQV onto EWT? 5. What congruence transformation maps NAV to BCY? 6. Is figure A congruent to figure B? Explain how you know. 7. Are the figures congruent? Explain how you know.

22 8. Verify the SAS Postulate by using congruence transformations. Given: J P, PA JO, FP SJ Prove: JOS PAF (Describe the rigid motions that map PAF onto JOS.) 9. Verify the SSS Postulate by using congruence transformations. Given: TD EN, YT SE, YD SN Prove: YDT SNE

23 Dilations Geometry Unit 9: Lesson 7 Name A dilation is a transformation that maps a figure onto a similar figure with respect to a center of dilation and a scale factor. Notation: DO,k (P ) = P Dilation of point P with center of dilation O and scale factor k. For DO,k (P ) = P P is the point on OP = k (OP ) such that OP P P O k(op ) For DO,2 (P ) = P P is the point on OP = 2 (OP ) such that OP P P O 2(OP ) Dilations preserve angle measurement. Dilations do not preserve distance. The distance changes by the scale factor. If k > 1, the dilation is an enlargement. If k < 1, the dilation is a reduction. If k = 1, the size does not change. The scale factor k of a dilation is the ratio of a length of the image to the corresponding length in the preimage. Therefore, a dilation is not a rigid motion or isometry. D C k, ( PRQ) ( P' R' Q') CR' k k CR CR' CR

24 C' D' k CD 4 k 2 k 2 Dilations in the Coordinate Plane If a dilation is centered at the origin DO,k (x, y) = (kx, ky) If a dilation is not centered at the origin D(a,b),k (x, y) = (a + k(x - a), b + k(y - b)) Notice a dilation takes a line not passing through the center of dilation to a parallel line and a line passing through the center unchanged. 1. a) A dilation is a transformation that results in an image that is to the original figure. b) The amount a figure is enlarged or reduced by a dilation is called the c) A dilation enlarges or reduces a figure with respect to a point called the, which may be located by drawing lines that connect points on the original figure to corresponding points on the image. d) If a line passes through the center of dilation, its image also passes through the. If a line does not pass through the center of dilation, its image is to the original line. 2. Which of the following properties are invariant under a dilation? a) distance (or betweeness) b) angle measure c) orientation d) perimeter e) area f) collinearity (points collinear on preimage are collinear on the image) g) parallelism (parallel lines/segments on preimage are parallel on the image) h) perpendicularity (perpendicular lines/segments on preimage are perpendicular on the image) 3. a) Is ( XTR) X ' T' ' an enlargement or reduction? D X, n R b) What is the scale factor n of the dilation?

25 4. a) Is D O, n ( JKLM ) J' K' L' M ' an enlargement or reduction? b) What is the scale factor n of the dilation? 5. a) PZG has vertices at P(2, -1), Z(-2, 1), and G(0, -2). Graph PZG and D ( PZG) P' Z' '. 2 G b) What do you notice about the slopes of corresponding sides? c) What does this tell you about corresponding sides under a dilation? 6. a) PZG has vertices at P(2, -1), Z(-2, 1), and G(0, -2). Graph PZG and D ( PZG) P' Z' '. 1 G 2 b) What do you notice about the slopes of corresponding sides? c) What does this tell you about corresponding sides under a dilation?

26 7. The height of a document on your computer screen is 20.4 cm. When you change the zoom setting on your screen from 100% to 25%, the new image of your document is a dilation of the previous image. a) What is the scale factor of this dilation? b) What is the height of the new image? 8. The line y = 2x 4 is dilated by a scale factor of 2 3 and centered at the origin. Which equation represents the image of the line after the dilation? a) y = 2x 4 b) y = 2x 6 c) y = 3x 4 d) y = 3x 6 9. The equation of line h is 2x + y = 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?

27 Similarity Transformations Geometry Unit 9: Lesson 8 Name Similarity Transformations Compositions of rigid motions and dilations can be used to understand similarity. Compositions of rigid motions and dilations map preimages to similar images. Therefore, they are called similarity transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure onto the other. 1. a) If two figures are similar, then there is a sequence of and a that map one figure onto the other. b) If there is a sequence of rigid motions and a dilation that map one figure onto the other, then the two figures are c) A sequence of rigid motions and a dilation is called a transformation. 2. a) DEF has vertices D(2, 0), E(1, 4), F(4, 2). 3. a) LMN has vertices L(-4, 2), M(-3, -3), N(-1, 1). What is the image of DEF when you apply Find the image of LMN under the following the composition D1.5 ry axis ( DEF )? composition D ( T 2 ( LMN)). b) Is this composition a similarity transformation? b) Is this composition a similarity transformation? 1 O, 2 4, 4. What is a composition of rigid motions 5. What is a composition of rigid motions and a

and a dilation that maps RST to PYZ? dilation that maps trapezoid ABCD to trapezoid MNHP? 6.

If so, identify the similarity transformation and write a similarity statement. If not, explain.

28 and a dilation that maps RST to PYZ? dilation that maps trapezoid ABCD to trapezoid MNHP? 6. Is there a similarity transformation that maps PAQ to TNO? If so, identify the similarity transformation and write a similarity statement. If not, explain. 7. Is there a similarity transformation that maps JKL to RST? If so, identify the similarity transformation and write a similarity statement. If not, explain. 8. A new company is using a computer program to design its logo. Are the two figures used in the logo similar? If they are similar, describe the similarity transformation.

Chapter 2: Transformations. Chapter 2 Transformations Page 1

Chapter 2: Transformations. Chapter 2 Transformations Page 1 Chapter 2: Transformations Chapter 2 Transformations Page 1 Unit 2: Vocabulary 1) transformation 2) pre-image 3) image 4) map(ping) 5) rigid motion (isometry) 6) orientation 7) line reflection 8) line