Reflections Day 1 A transformation is a function that moves or changes a figure in some way to produce a new figure call an image. Another name for the original figure is the pre-image. We will study the following transformations: Reflections Translations Rotations Dilations R Resizing! 1
Vocabulary to know: A Rigid Motion is a transformation that preserves length (distance preserving) and angle measure (angle preserving). Another name for a rigid motion is an isometry. Orientation refers to the arrangement of points relevant to one another, after a transformation has occurred. A direct isometry preserves distance and orientation. Translations and Rotations are direct isometries. An opposite isometry preserves distance but does not preserve orientation. Line reflections are opposite isometries. 2
Example 1: Example 2: Example 3: Match each of the following 1) Rotation a) 2) Translation b) 3) Line Reflection c) 4) Dilation d) 3
Line Reflections Line reflections are isometries that do not preserve orientation. Therefore line reflections are opposite isometries. Thus, a reflected image in a mirror appears "backwards." A line reflection is a rigid motion. Example: a. 4
Reflections in the Coordinate Plane 1. 3. 4. 5
5. Triangle: A(1,2), B(1,4) and C(3,3). Reflection: Origin A ( ) B ( ) C ( ) Rule: r origin (x,y) = (, ) Examples: 1. What is the image of point A(4,-7) after a reflection in the x-axis? 2. What is the image of point B(2,-3) after a reflection in the y-axis? 3. Find the coordinates of C, the image of C(5,-2), after a reflection in r y=x. 4. Find the coordinates of the image of point D(-3,7) after a reflection in the line y = -x. Constructions and Line Reflections. Example 1: is reflected across and maps onto. Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting to, to, and to. What do you notice about these perpendicular bisectors? Label the point at which intersects as point. What is true about and? How do you know this is true? 6
Example 2: Construct the segment that represents the line of reflection for quadrilateral and its image. What is true about each point on and its corresponding point on? Examples 3 4 Construct the line of reflection across which each image below was reflected. 3. 4. You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore methods for constructing the reflected image itself. The first few steps are provided for you in this next stage. 7
Example 5 The task at hand is to construct the reflection of over line. Follow the steps below to get started, then complete the construction on your own. 1. Construct circle : center, with radius such that the circle crosses at two points (labeled and ). 2. Construct circle : center, radius and circle : center, radius. Label the [unlabeled] point of intersection between circles and as point. This is the reflection of vertex across. 3. Repeat steps 1 and 2 for vertices and to locate and. 4. Connect,, and to construct the reflected triangle. Things to consider: When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection itself? Why did the construction above work? Example 6 Now try a slightly more complex figure. Reflect across line. 8
Reflecting Over Vertical and Horizontal Lines 1. 2. 9
Identifying a Reflection Write a rule to describe each transformation. 1. 2. Challenge 10
Summary Exit Ticket Homework: Homework for Reflections Day 1 11
Homework 1. Find the reflection of the point (5,7) under a reflection in the x-axis. 2. Find the reflection of the point (6,-2) under a reflection in the y-axis. 3. Find the reflection of the point (-1,-6) under a reflection in the line y =x. 4. 5. 12
Graphing Reflections 13
Identifying Reflections 14
Construct the line of reflection for each pair of figures below. 1. 2. 3. 4. Reflect the given image across the line of reflection provided. 15
5. Draw a triangle. Draw a line through vertex so that it intersects the triangle at more than just the vertex. Construct the reflection across. 16
Translations Day 2 Do Now: 1. 2. A translation is a rigid motion. A translation is a direct isometry. It preserves distance and orientation. 17
a. b. What is the magnitude of the vector? 18
Example 5: a) Name the vector and write its component form. b) What is the magnitude of the vector? 19
Practice Problems 1. The vertices of ΔLMN are L(2,2), M(5,3) and N(9,1). Translate ΔLMN using the vector <-2, 6 >. 2. Graph ΔRST with vertices R(2,2), S(5,2) and T(3,5) and its image under the translation (x,y) ( x+ 1, y + 2). 3. Δ DEF is a translation of ΔABC. Use the diagram to write a rule for the translation of ΔABC to ΔDEF. 20
4. 5. 6. a) Write a motion rule for the translation. b) What is the magnitude of the vector? 21
7. Describe the composition of transformations shown in the diagram. 8. 22
Homework 1. 2. 3. a) Write a motion rule for the translation. b) What is the magnitude of the vector? 23
4. Write a rule for the translation. 5. Find the component form of the vector that translates P(-3,6) to P (0,1). 6. Describe the composition of transformations shown in the diagram. 7. 24
Rotations Day 3 Warm - Up 1. 2. A rotation is a rigid motion. 25
1a. Rules (x, y) R 90 = (, ) R 180 = (, ) R 270 = (, ) Practice (2, 3) (-1, 6) (-7, -4) 3. 4. 5. 26
6. Given triangle X(2,2), Y(4,3) and Z(3,0). Write the coordinates of its image after a rotation 90 0. Rule: (x,y) ( ) X ( ) Y ( ) Z ( ) 7. Given triangle A(1,3), B(4,4) and C(4,0). Write the coordinates of its image after a rotation 180 0. Rule: (x,y) ( ) A ( ) B ( ) C ( ) 8. Given triangle A(-2,-5), B(-1,1) and C(1,-2). Write the coordinates of its image after a rotation 270 0. Rule: (x,y) ( ) A ( ) B ( ) C ( ) 27
Finding the center of rotation using constructions 1. Follow the directions below to locate the center of rotation taking the figure at the top right to its image at the bottom left. a. Draw a segment connecting points and. b. Using a compass and straightedge, find the perpendicular bisector of this segment. c. Draw a segment connecting points and. d. Find the perpendicular bisector of this segment. e. The point of intersection of the two perpendicular bisectors is the center of rotation. Label this point. 2. Find the center of rotation: 28
Identifying a Rotation Write a rule to describe each transformation. a. b. Challenge: 29
Homework: 1. Graph the image of the figure under the transformation given and write the coordinates of the image. a) b) c) 30
2. Write the rule for the transformation. 3. Write the rule for the transformation. 4. Graph the image of triangle DEF after a rotation of 180 o. 31
5. Use a compass and a straightedge to find the center of rotation. 6. On your paper, construct an equilateral triangle. Locate the midpoint of one side using your compass. Rotate the triangle around this midpoint. What figure have you formed? 7. Write the rule for the transformation. 32
Dilations/Symmetry Day 4 Warm Up 1. 2. 33
******CHOOSE AN EXPLANATION A dilation is NOT a rigid motion. *It does not preserve distance and is not an isometry. A dilation preserves: 1) Angle measure 2) Parallelism 3) Colinearity 4) Midpoint 5) orientation 34
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Example: Use a compass and straightedge to construct the dilation of scale factor 3. LMN with the given center and 37
Homework: 1. 2. 3. 4. 38
5. Graph B(-5,-10), C(-5,0) and D(0,5) and its image after a dilation of -1/5. 6. 39
7. Find the scale factor of the dilation for each of the following: a) b) c) 8. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor ½. 40
9. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor 2. 41
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Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry. 43
Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry. 44
Summary: 45
Compositions and Rigid Motions Day 5 Warm Up 1. 2. Which rotation about point O maps B to D? A composition of transformations is when two or more transformations are combined to find a single transformation. The composition of two or more rigid motions is a rigid motion. 46
Example 1: The symbol for a composition of transformations is an open circle. The notation is read as a reflection in the x-axis following a translation of (x+3, y+4). Be careful!!! The process is done in reverse!! You may see various notations which represent a composition of transformations: could also be indicated by 47
Example 2: 48
Example 3: 49
Example 4: 50
Example 5: Use the figure. The distance between line k and line m is 1.6 centimeters. a) The preimage is reflected in line k, then in line m. Describe a single transformation that maps the figure on the far left to the figure on the far right. b) What is the relationship between PP and line k? Explain. c) What is the distance between P and P? 51
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Homework: 1. Identify any congruent figures in the coordinate plane. 2. Describe a congruence transformation that maps Quadrilateral SPQR to ZWXY. 3. Describe a congruence transformation that maps triangle ABC to triangle EFG. 4. a) Evaluate the composition of r x axis o R 90 (-3,5). b) What single transformation would be the same as the composition in part a? 54
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9. Determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning. a) b) W(-3,1), X(2,1), Y(4,-4), Z(-5,-4) and C(-1,-3), D(-1,2), E(4,4), F(4,-5) 57
10. Find the angle of rotation that maps A to A. 11. 58
Test Prep Questions 1. Which transformation preserves both distance and angle measure? a) Translation b) horizontal stretch c) vertical stretch d) dilation 2. Which description is always the composition of two different types of transformations? a) Line reflection b) rotation c) translation d) glide reflection 3. Which of the following transformations preserve angle measure but not distance? a) Dilation b) translation c) vertical stretch d) rotation 4. AB has a length of 2 10. What is the length of A B, the image of AB after a dilation with a scale factor of 3? a) 5 10 b) 2 30 c) 6 10 d ) 6 30 5. Which transformation is represented in the table below? a) T -1,3 b) T 1,3 c) T -3,1 d) T 1,-3 Input Output (3,5) (4,2) (1,0) (2,-3) (6,3) (7,0) 6. Which rigid motion does not preserve orientation? a) line reflection b) rotation c) translation d) dilation 7. Which transformation is not considered a rigid motion? a) dilation b) rotation c) reflection d) translation 8. Which transformation is equivalent to a reflection over a point? o o a) T (2,5) b) R 90 c)r 180 d) r x-axis 9. Which of the following composition of transformations maps ABC to A B C? 1) R 180 o o T -2,2 2) T 1,4 o r y-axis 3) T 3,-2 o r x-axis 4) R 90 o o r y=x 59
10. A series of three transformations that, when performed on ABC, finally resulting in A B C. Identify each of the three transformations: a) ABC to A B C b) A B C to A B C c) A B C to A B C d) Is ABC A B C? Justify your answer. e) Identify a single transformation that, when performed on A B C, would result in a triangle A B C such that ABC is not congruent to A B C. Justify your answer. 11. Use a sequence of rigid motions to show that 1 2. Use coordinate notation. Label the vertices in triangle 2 using A B C to correspond to the appropriate angles in triangle 1. 60
12. a) On the accompanying grid, graph and label XYZ with coordinates X(0,2), Y(5,0) and Z(1,3). b) On the same grid graph and label X Y Z, the image of XYZ after the composition of transformations R 180 o o r x-axis. c) What single transformation maps XYZ to X Y Z? 13. Given circle A with the equation (x-2) 2 + (y + 3) 2 = 4. a) Perform the translation of circle A: T(2,-4) and label the translation as circle A b) Perform a dilation of circle A about its center with a scale factor of 2. Label the image A. c) Write the equation of circle A and circle A. 61
Lesson Pages in Packet Homework Assignment Day 1 Reflections 1 16 12-16 Day 2 Translations 17-24 23-24 Day 3 Rotations 25 32 30 32 Day 4 Dilations/Symmetry 33-45 38 41 and page 44 Day 5 Compositions and Rigid 46-58 54-58 Motions Day 6 Test Prep Questions 59-61 59-61 62