Transformations & Basic Geometry Vocabulary

Transcription

1 Transformations & Basic Geometry Vocabulary

2 Key Terms and Concepts

3 Name: Date: Period: Rigid Transformations Toolkit Key Vocabulary- Give a precise definition of each term Parallel lines: Perpendicular lines: Circle: Rigid transformation: Non-Rigid Transformation: Rotation: Reflection: Translation: Dilation: Construct the perpendicular bisector of AB Construct a line perpendicular to AB through P Construct a line perpendicular to AB through P A A A P B P B B IMP Toolkit: Rigid Transformations 1

4 Compare the characteristics of the three rigid transformations Characteristics of Translations Characteristics of Reflections Characteristics of Rotations Determine the symmetries of each figure Lines of Symmetry(Y/N) Number of Lines of Symmetry Rotational Symmetry(Y/N) Order of the rotational symmetry Angle of rotation Rectangle Rhombus Parallelogram Equilateral Triangle Square Regular Pentagon Regular Hexagon IMP Toolkit: Rigid Transformations 2

5 Mark the congruent and parallel parts on each transformation. Rotation: Find the coordinates of A and A Write the rule of rotation in coordinate notation: What is the center of rotation? What is the angle of rotation? Reflection: Draw the line of reflection. A Describe how you find the line of reflection: C 1 B 1 A 1 Translation: A Draw the translation vector. B Write the rule of translation in coordinate notation: C A 1 C 1 B 1 IMP Toolkit: Rigid Transformations 3

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7 Math I Transformations and Geometric Vocabulary Storybook

8 The Math Behind Essential Question # 1 Which construction(s) are necessary to perform each transformation? Essential Question # 2 Which transformations preserve congruence? Essential Question # 3 What is similar and different about rigid and non-rigid motions? Coordinate Notation Coordinate Notation is used to describe transformations as functions that take points in the plane as inputs and give other points as outputs. The general point, (x, y) is used to represent a point in the pre-image. If the input is translated right or left by c units, the x value of the point will increase or decrease by c units and if the input is translated vertically d units, the y value of the input will increase by d units; e.g. (x,y) x+c, y+d). The following table summarizes key transformations represented in words and coordinate notation. Transformation In Words Coordinate Notation Translation Move each point left or right c units and up or down d units (x, y) (x + c, y + d) Reflection Reflection across the x-axis (x, y) (x, - y) Reflection Reflection across the y-axis (x, y) (- x, y) Reflection Reflection across the line y = x (x, y) (y, x) Rotation Rotation 90 clockwise about the origin (x, y) (y, - x) Rotation Rotation 180 about the origin (x, y) (- x, - y) Rotation Rotation 270 about the origin (x, y) (-y, x) Dilation Enlargement or reduction with by a scale factor of k (x, y) (kx, ky) Notation for the pre-image and image In describing transformations, we label the original figure with lower case letters and refer to it as the pre-image. The figure created as a result of transformations is referred to as the image and the points are labeled with the same letters but prime. For example x becomes x. If x is then transformed, the new point is labeled x. Constructions: Copying a Segment To copy a segment, the compass is opened to the distance of the segment to be copied. A point is drawn to represent the end point of the copied segment and then the compass is used to draw the other end point of the segment. A straightedge can then be used to draw the segment. Drawing a Circle To draw a circle with a compass, set the compass to be the width of the radius of the desired circle. Mark the center of the circle with a pencil and then place the point of the compass on the center. With the compass extended the distance of the radius, leave the point of the compass on the center and trace around until the pencil returns back to the beginning. Constructing an Equilateral Triangle While this can be done multiple ways, the unit leads students to the following process. Draw a horizontal line segment the desired length of a side of the equilateral triangle (below, segment PQ). Keeping the compass at the same width, place the point of the compass on the left endpoint of the segment (P) and draw an arc (part of a circle) above and to the right. Now place the compass on the right endpoint of the segment (Q) and make an arc above and to the left of this endpoint. The point where the two arcs intersect (R) is the third vertex for the triangle, as each arc represents points equidistant from each vertex. Rigid vs. Non-Rigid Motion Rigid Motion preserves congruence. The transformations that are rigid motions are translations, rotations and reflections. Rigid motions preserve distance as well as angle

9 The Unit! measure and thus shapes transformed by a rigid motion remain congruent. Non-Rigid Motion preserves similarity but not congruence. Dilations are non-rigid motions. Non-rigid motions maintain angle measure but distances in the image are proportional to distances in the pre-image. Dilations: When a figure is dilated (about the origin), each point on the pre-image is multiplied by the scale factor to result in the image. Figures can be dilated by scale factors greater than 1 (enlargements) or between 0 and 1 (compressions). On a coordinate plane, the dilated sides maintain the same slope as the corresponding sides on the pre-image and thus, the angle measures remain congruent. The side lengths increase or decrease proportionally by the scale factor and thus dilations produce similar figures. When a figure is dilated about a point other than the origin, each point in the figure is moved along the line segment connecting it to the given point to a point whose distance from the given point is a given multiple (the scale factor) of the original distance of the point on the figure to the given point. Point of Dilation Figures can be dilated about a point outside the figure, inside the figure or on the figure. The point from which the figure is dilated affects the location of the dilated image, but not angle measures or distances. Scale Factor The factor by which each coordinate is multiplied is called the scale factor. The scale factor is the ratio of the side lengths in dilated figures. Dilation by Construction Off the coordinate plane, to dilate a figure we must first be given the point of dilation and the scale factor. We then draw a ray from the point of dilation through each vertex of the pre-image. Using the compass, we now copy the segment from the point of dilation (in the case below, c) to the vertex of the pre-image as many times as the scale factor of the dilation (in the case below, two times) along the ray. We repeat this process for each of the other rays and then connect the vertices of the image. Rotations: Rotations are rigid motions in which a point or figure is rotated about a point (the center of rotation) a given number of degrees (angle of rotation). The resulting image is congruent to the original figure as rotations preserve distance and angle measure. Center of Rotation The Center of Rotation is the point about which the figure is rotated. The center of rotation can be the origin, a point on the figure, a point outside the figure or a point inside the figure. Angle of Rotation The angle to rotation is the angle through which the figure will be rotated (from 0 to 360 degrees). It is common to describe the direction of the rotation (clockwise or counter clockwise) Determining the Angle of Rotation When given an image and its pre-image, to determine the angle of rotation, we first must know the center of rotation. Given this point, we draw lines connecting the center of rotation to a corresponding vertex on the image and pre-image. These line segments now form the angle of rotation, which can be measured with a protractor. In the example below, D represents the center of rotation; line segments have been draw connecting D to both A and A. The angle formed by A DA is the angle of rotation, in this case, 90 counterclockwise. Reflections: Reflections are rigid motions in which a point or figure is reflected (flipped) across a given line, called the line of reflection. Reflections preserve congruence. Academic Language Reflection Line of reflection Rotation Point of rotation Translation Rigid transformation Dilation Perpendicular bisector Compass Protractor Angle Circle Parallel Perpendicular Line segment Pre- image Coordinate notation Symmetric Distance Congruence Rotational symmetry Order of rotational symmetry Center of dilation Real-World Application Video Games Movie Animation Photo enlarging Ferris Wheel and other Ride Construction Bridge construction Scale Models/ Drawings

10 Math Practice Standards 1) Make sense of problems and persevere in solving them. The Math Behind Line of Reflection The line of reflection is the line across which the point(s) or image is reflected. The line of reflection can be an axis on the coordinate plane or any line. Corresponding points on the image and pre-image are equidistant from the line of reflection. The line of reflection is the perpendicular bisector of each segment connecting corresponding points of the image and pre-image. Reflections by Construction The following are the steps taken to reflect a point P over a given line, AB. 2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the reasoning of others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precision. 7) Look for and make use of structure. 8) Look for and express regularity in repeated reasoning. To find the line of reflections given a point and its pre-image, we adjust the compass to be a distance greater than half the length of the segment connecting the image to its pre-image. We set the compass on the image and make an arc both above and below the point (or draw a full circle). We repeat this process for the preimage point. There will be two places where the arcs intersect, now representing two points on the line of reflection. Translations Translations are rigid motions in which a point or figure moves vertically and/or horizontally a given number of units. Translations preserve congruence and are defined by the direction and number of units of the translation or by using translation vectors. Translation Vectors Translations vectors can be used to explain how to translate a figure or point. A single vector can represent both a horizontal and a vertical translation. On the coordinate plane, when a pre-image is translated according to the translation vector, each point moves both the vertical and horizontal distance represented by the slope of the vector. The segments connecting the corresponding points on the image and pre-image are parallel to the translation vector (as they all have the same slope) are the same length as the translation vector. Common Congruence A common misconception is that congruent figures must look exactly the same upon first glance. If congruence is first defined as same size and same shape, it can be hard for students to recognize congruent figures in which the preimage has undergone a series of transformations, such as the one shown below. If students come to define congruence as a figure that maps onto itself after a series of translations, reflections and rotations, they will avoid this misconception. Similarity A common issue is the oversimplification of defining similar figures as having the same shape but a different size. This definition is imprecise and leads to further misconceptions and hinders the approach to proof in Math II. Similarity should be defined in terms of a series of translations, reflections or rotations and a dilation. Rather than trying to observe or measure figures to determine if they seem to be the same shape, students need to perform transformations to show similarity. In addition, students come to define similarity as a result of studying the effects of dilations on figures.

11 The Unit! (Cont.) Translations by Constructions Following are the steps that can be taken to translate a figure by a given translation vector DE. Repeat the steps for points B and C to find B and C. Symmetry Line of Symmetry A line that reflects a figure onto itself is called a line of symmetry. Ex: The dotted line is a line of symmetry in an isosceles triangle. Rotational Symmetry A figure that can be carried onto itself by a rotation is said to have rotational symmetry. As seen in the example of the Ferris Wheel, each time the wheel rotates 30, the figure will look identical to the pre-image (the seats will all be in the same location). Thus, this 12-sided regular polygon has rotational symmetry. Order of Rotational Symmetry and Angle of Rotation The number of times a figure can be rotated onto itself until a given point on the figure returns to its original position is called the order of rotational symmetry. In the example of the Ferris Wheel, the dodecagon will rotate onto itself 12 times until a given point returns to its original position and thus this figure has an order of rotational symmetry of 12. The Angle of Rotation is the angle of the smallest turn until the figure rotates back onto itself. In the example above, the figure will rotate onto itself every 30. Symmetry in Polygons Regular polygons (meaning all side lengths are congruent and all angle measures are congruent) will have as many lines of symmetry as it has sides, and the order of rotational symmetry will equal the number of sides. The angle of rotation can be determined by dividing 360 by the number of sides in a regular polygon. For example a regular octagon (stop sign) has 8 lines of reflection, rotational symmetry order of 8 and the angle of rotation is (360 8) or 45. Misconceptions Transformations are isolated to the world of geometry It is easy to be narrow in our focus and see transformations solely as part of the study of geometry. It is better to see them in the larger category of functions that take points in the plane as inputs and give other points as outputs. Using coordinate notation to describe transformations helps students make the connection between transformations and functions. Equilateral Triangle Many students focus on the equal side lengths that comprise an equilateral triangle and thus, when trying to construct one, may only rely upon a ruler/straightedge. In allowing students to try to construct an equilateral triangle using only a straightedge, they will come to understand that the measures of the angles must also be precise. This struggle will also help students understand the usefulness of a compass in constructions and in helping understand definitions. Rotations Rotations can be easy to recognize, but students need to learn precision in terms of defining rotation in terms of both the center of rotation and the angle of rotation. As transformations will be used as the primary means by which to prove figures to be congruent or similar in future units, students need to learn early to define rotations precisely. Student Talk Strategies Report to a partner Each student reports his/ her own answer to a peer. The students listen to their partner s response. ( Turn to a partner on your left. Now turn to a partner on your right etc.) Give one get one After brainstorming ideas, students circulate among other students sharing one idea and getting one. Students fold paper lengthwise they label the left side give one and the right side get one. Think, Pair, Share Students think about a topic suggested by the teacher. Pairs discuss the topic. Students individually share information from their discussion with the class. Inside-outside circle Two concentric circles of students stand or sit, facing one another. The teacher poses a question to the class, and the partner responds. At a signal, the outer of inner circle or outer circle rotates and the conversation continues. Appointment clock Partnering to make future discussion/work appointments. Jigsaw Group of students assigned a portion of a text; teach that portion to the remainder of the class.

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13 1 Objective: Students will draw on their prior knowledge of translations and rotations to fit pieces together in the Tetris board. TASK A The game of Tetris is a video game of pieces made of four squares in different arrangements. There are seven different shapes that could be given the player- the I, S, J, L, Z, O and T. These shapes can be rotated clockwise or counter clockwise and moved (translated) side to side. They cannot be flipped over (reflected). The goal of the game is to make complete rows with the pieces so that they drop out. The seven shapes are below. Rotate each of the shapes counter-clockwise in 90 intervals and draw them in the boxes to the right. Name Beginning block Rotated 90 Rotate 180 Rotate 270 I Shape Z Shape 1 J shape 1 O Shape 1 L Shape 1 T Shape 1 S Shape 1 IMP Activity Tetris 1

14 TASK B Your brother is very competitive and has challenged you to a Tetris battle. He began the following game and has given it to you to finish. Your partner will give you the next ten shapes in the order of their choice. Describe how you would change the shape (units to the right/ left and rotation) and give the resultant number of rows. Pieces enter the game board on the arrow. Next ten pieces 1. Transformation (rotation and translate right/ left) Define ROTATION: Define TRANSLATION: IMP Activity: Tetris 2

15 TASK C 1) Which order did you give the pieces to your partner? Why did you choose this order? 2) What improvements could you make in the order you gave your partner pieces to prevent them from making complete lines? Explain. 3) Which of the pieces rotates onto itself? Explain. 4) Describe how you can rotate each piece to get it back to itself. O Piece I Piece S Piece Z Piece T Piece L Piece J Piece 5) How did translating the pieces left or right change the shape? What was the same? What was different? 6) How did rotating the pieces change the shape? What was the same? What was different? IMP Activity: Tetris 3

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19 Objective: Students review and practice coordinate notation. Students also practice translating, rotating and dilating points on a coordinate grid. Introduction to Coordinate Notation for Transformations TASK A Transformation In words Coordinate Notation Translation Move each point left or right c units and up or down d units (x, y) (x + c, y + d) Example : (2, 3) (in words) A) Translate (2, 3) 6 units down and 3 units right. Resulting coordinates of new point Reflection Reflection Reflection Rotation Rotation Rotation Dilation Reflection across the x-axis Reflection across the y-axis Reflection across the line y = x Rotation 90 clockwise about the origin Rotation 180 about the origin Rotation 270 clockwise about the origin Enlargement or reduction with by a scale factor of k (x, y) (x, y) (x, y) ( x, y) (x, y) (y, x) (x, y) (y, x) (x, y) ( x, y) (x, y) ( y, x) (x, y) (kx, ky) B) Reflect (2, 3) across the x-axis C) Reflect (2, 3) across the y-axis D) Reflect (2, 3) across y = x. E) Rotate (2, 3) 90 CW about the origin F) Rotate (2, 3) 180 about the origin G) Rotate (2, 3) 270 clockwise about the origin H) Dilate (2, 3) by a factor of 2. Create a 10 x 10 coordinate plane on the grid to the right. Plot each of the example points from the chart. Label each point with the corresponding letter from the description in the chart above. IMP Pre-Unit: Introduction to Coordinate Notation for Transformations 1

20 TASK B: Directions: On the grid below create an 8x8 coordinate plane. Plot the point A(0,2), then create new points according to the following transformations. The coordinates for Point B are found by transforming point A. The coordinates for Point C are found by transforming point B. Complete the table using this pattern to find the remaining points. Important: Label each point. DO NOT connect the points until the very end. Directions for connecting the points are given below the table. Transformation New Coordinates 1. Translate A up 5 and right 3 B (, ) 2. Reflect B over the y-axis C (, ) 3. Translate C down 10 D (, ) 4. Reflect D over x-axis E (, ) 5. Rotate E 180 about origin F (, ) 6. Reflect F over x-axis G (, ) 7. Translate G (x 2, y 1) H (, ) 8. Reflect H over y-axis I (, ) 9. Translate I (x + 4, y 1) J (, ) 10. Reflect J over y-axis K (, ) 11. K(x, y) (x 1, y 2) L (, ) 12. Translate L up 1 and right 7 N (, ) 13. Rotate N 90 clockwise around the origin. O (, ) 14. O(x, y) (x, y + 6) P (, ) Connect the Points in this order: A, B, G, H, J, N, F, O, D, L, K, I, E, C, P Optional additional points 1. Plot and connect the following points in order. Also, connect the first point to the last point. ( -2, 1 ), ( -1, 1 ), ( -1, 0) 2. Plot and connect the following points in order. Be sure to connect the first point to the last point. ( 2,1 ), ( 1, 1), ( 1, 0) 3. Plot and connect the following points in order. Be sure to connect the first point to the last point. ( -1, -1), ( 1, -1 ), ( 0, -2) 4. Plot and connect the following points in order. DO NOT connect the first and last point. ( -3, -2), ( -2, -3 ), ( -1, -3), ( 0, -2), ( 1, -3), ( 2, -3), ( 3, -2) IMP Pre-Unit: Introduction to Coordinate Notation for Transformations 2

21 Name Date Period How did that shape move? Directions: For each grid with figures, use patty paper to trace the original figure, then complete the chart and answer the questions below. TASK A: Transformation #1: (-3,5) Original Image B (-3,1) Image C How was the shaped moved (transformed) to make images B and C? Original Coordinates (, ) (, ) Coordinates for Image B Coordinates for Image C What math (operation and amount) was done to the original coordinates to transform the original shape to images B and C? Are images B and C the same, congruent, similar or none of these compared to the original image? Explain your choice of word(s). Summary: The surfboard was translated 6 units right for Image B and 6 units down for Image C. What does it mean to translate a shape and how does this affect the coordinates and the size and shape of the figure? 1 IMP Activity: How Did That Shape Move?

22 Transformation #2: Original Image B Image C How was the shaped moved (transformed)? Original Coordinates Coordinates for Image B Coordinates for Image C What math (operation and amount) was done to the original coordinates to transform the original shape to images B and C? Are images B and C the same, congruent, similar or none of these compared to the original image? Explain your choice of word(s). Summary: The surfboard was reflected across the line x=0 for Image B and reflected across the line y=0 for Image C. What does it mean to reflect a shape and how does this affect the coordinates and the size and shape of the figure? 2 IMP Activity: How Did That Shape Move?

23 Transformation #3: Original Image B Image C How was the shaped moved (transformed)? Original Coordinates Coordinates for Image B Coordinates for Image C What math (operation and amount) was done to the original coordinates to transform the original shape to images B and C? Are images B and C the same, congruent, similar or none of these compared to the original image? Explain your choice of word(s). 3 IMP Activity: How Did That Shape Move?

24 Summary: The surfboard was dilated by a factor of 2 for Image B and dilated by a factor of 0.5 for Image C. What does it mean to dilate a shape and how does this affect the coordinates and the size and shape of the figure? Respond to either one of these two prompts: What idea or ideas do you understand better as a result of doing today s activities? What are three mathematical terms that were used today and what do they mean? List one or two questions that you have either about today s work or about the topic of transformations in general. IMP Activity: How Did That Shape Move? 4

25 Name Date Class Transformations Review Use the triangles for 1 5. Use the triangle for Which triangle is a translation of the gray triangle? A ABC B DEF C neither 2. DEF is translated 5 units straight up so that point F lands on the x-axis. Where does point D land? A (0, 5) B (5, 0) C (5, 5) 3. The translation below is used on all three triangles. (x, y) (x, y 1) What happens to the triangles? A They all move 1 unit to the right. B They all move 1 unit to the left. C They all move 1 unit up. 4. Which triangle is a reflection of the gray triangle? A ABC B DEF C neither 5. Which triangle has the same orientation as the gray triangle? A ABC B DEF C neither 6. The triangle is rotated 90 (one-quarter turn). Point Q is used as the center of this rotation. Which describes the new position of side PQ? A vertical B horizontal C slanted 7. The triangle is rotated 180 with point Q as the center of rotation. What is the new position of point P? A ( 2, 1) B (1, 4) C (4, 1) 8. The translation below is used on the triangle. (x, y) (x 4, y 4) What is the new location of point R? A (1, 5) B (5, 1) C (4, 4) PQR is dilated by a scale factor of 2. The center of the dilation is (0, 0). 9. How does the image of triangle PQR relate to triangle PQR? A It is smaller. C It is the same size. B It is larger. 10. The new location of point P is ( 4, 2). What is the new location of point Q? A (2, 2) B (2, 4) C (4, 4) 11. Triangle PQR and its image are the same shape but not the same size. What word do you use to describe this? A congruent B dilated C similar Original content Copyright by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 1

26 Name Date Class Transformations: Review Use the triangles for Translate DEF 1 unit to the right. Write the ordered pairs for the new vertices.,, 18. Reflect LMN across the y-axis. 19. Rotate LMN 90 clockwise about the origin. 20. Rotate LMN 180 about the origin. Use the figure for Reflect DEF across the x-axis. Write the ordered pairs for the new vertices.,, 14. The gray triangle is a 90 rotation of PQR. Is the direction of rotation clockwise or counterclockwise? 15. Apply the translation below to DEF. Write the ordered pairs for the new vertices. (x, y) (x, y 4) 21. Dilate the rectangle using a scale factor of 2. Use the origin as the center. Draw your answer on the grid. 22. How did the length and width change? 23. Label the new rectangle W', X', Y', Z'. Write the ratio of WX to W'X'.,, 16. Apply the reflection below to PQR. Write the ordered pairs for the new vertices. (x, y) ( x, y),, to Compare it to rectangle WXYZ. 24. Compare the small and large rectangles. Their areas are different, but what stays the same? 17. Describe a translation to PQR so that point P lands on top of point D. Use the grid for Draw your answers on the grid and label each one with the problem number. Original content Copyright by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 2

27 Name Date Class Transformations Review Performance Task Designing with Transformations This design was completely created using only transformations of the black quadrilateral. In this activity, you will analyze this design to find out how it was made. Then you will use transformations to create a design of your own. You Will Need graph paper a ruler colored pencils Start by solving these problems. 1. The top three figures in the first quadrant were made with translations. Describe how these were made in words. 2. Describe the three first-quadrant translations using the (x, y) notation. 3. The three figures around the origin were created with rotations. Describe the three rotations in words. 4. Describe the three rotations using the (x, y) notation. 5. The two large figures were made with a dilation, a translation, and a reflection. Describe how these were made. Original content Copyright by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 3

28 Name Date Class 6. Now create a design of your own on a separate sheet of paper. Use all four types of transformations: translations, reflections, rotations, and dilations. Record the transformations you use. Original content Copyright by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 4

29 Name: Date: Period: Compass 1) Practice using your compass. Make two, different-sized circles in the space below. 2) Construct a circle centered at A and with a radius of AB.

30 3) Copy segment AB onto this line. Label the endpoints A and B. 4) Follow the directions to answer the question. a) Draw a circle. b) Copy that circle, placing the center somewhere on the circle. c) Continue to copy that circle placing the center on the intersection of the original circle and the copied circles. d) Connect the intersections on the original circle. What shape did you make?

31 -1- x t2q0a1j2a WKuu2t5a0 pssonfqtgwta2rwed ZLzLACx.l t KASljlO frpiagwhstcsx Lr Oeisje6rBvjeIdd.G t zmjagdrep lwci7tdhw eiynjfxijnzistfey YGyeeoNmrertFrny3.9 Worksheet by Kuta Software LLC Name Line Segment Constructions Construct a line segment congruent to each given line segment. 1) Date Period Construct a line segment whose length is equal to the sum of the lengths of the given line segments. 2) Construct a line segment whose length is equal to the difference of the lengths of the given line segments. 3) Construct a line segment the given number of times longer than the given segment. 4) 2 times as long Construct a line segment half as long as the given line segment. 5)

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33 Objective: Students will attempt to make an equilateral triangle with a ruler, then construct an equilateral triangle while practicing copying a segment and constructing congruent circles. Constructing an Equilateral Triangle TASK A Method 1: Using just a ruler, make an equilateral triangle. You may choose the size. Use a sheet of patty paper to check if your triangle is equilateral. You can trace the triangle and the tracing should fit your triangle no matter how you turn or flip the patty paper. Describe your strategy and what, if anything, was difficult. Method 2: Using a compass and a straightedge, make an equilateral triangle. You may choose the size. Use a sheet of patty paper to check if your triangle is equilateral. You can trace the triangle and the tracing should fit your triangle no matter how you turn or flip the patty paper. Describe your strategy and what, if anything, was difficult. IMP Activity: Constructing an Equilateral Triangle 1

34 TASK B: After the class discussion of page 1, use a compass and straightedge to construct equilateral triangles on each of the segments below. List the steps needed to make an equilateral triangle with a compass and straight edge. Explain how you know that the triangles you made above are equilateral. IMP Activity: Constructing an Equilateral Triangle 2

35 Objective: Students will determine the difference between rigid and non-rigid transformations. Students learn about scale factor and define dilation. They also learn to construct dilations from different centers of dilation. Rigid Motion vs Non-rigid Motion Transformations What are some things you think of when we say going to a theme park. Have you ever been to a theme park? If so, when and where was it? What was your best memory about the theme park? If not, where would be the first theme park you would like to go to and why? Rigid Motion Transformations: A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions preserve distances and angle measures. Referring to the virtual Theme Park: Describe the object, identify the transformation, describe the transformation and state if it is a rigid motion transformation or not. Object 1: Object 2: Object 3: Object 4: A dilation is another type of transformation. Why was it not listed in Rigid Motion Transformations? IMP Lesson: Rigid vs. Non-Rigid Transformations 1

36 TASK A: 1. Use the rule at the top of each column to find the coordinates of each new image. Plot each new triangle on the graph. (x,y) (2x,2y) (3x,3y) (4x,4y) (5x,5y) A(1,2) A (, ) A (, ) A (, ) A (, ) B(3,1) B (, ) B (, ) B (, ) B (, ) C(3, 3) C (, ) C (, ) C (, ) C (, ) 2. Explain what happened to Triangle ABC. 3. How much larger is the length of A B than the length of AB? 4. How much larger is the length of A B than the length of AB? 5. How much larger is the length of A B than the length of AB? IMP Lesson: Rigid vs. Non-Rigid Transformations 2

37 6. How are lengths related to the factor by which you multiplied the original coordinates? 7. Find the slopes of each of the following segments. a) AB b) A B c) A B d) A B e) A B 8. What conclusion can you draw from your answers to question 7? 9. Draw a line through all the A s, then another line through all the B s and then another line through all the C s. Make sure your line extends across the entire coordinate plane. What do you notice? The point where all the lines intersect is called the center of dilation. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. Label the center of dilation X. 10. How much larger is the length of A X than the length of AX? 11. How much larger is the length of A X than the length of AX? 12. How much larger is the length of A X than the length of AX? 13. Verify your answers to questions using a compass. Definition of Dilation: A Dilation about a point, A, of magnitude k is a transformation that sends the point X to the point X' lying on the ray determined by AX at distance k times the distance from A to X.. The point A is called the center of dilation. Each point is moved a distance from the center of dilation by the factor k, called the scale factor. A geometric figure in the plane subjected to a dilation will be similar to the original figure, but not congruent unless k= Define dilation in your own words. IMP Lesson: Rigid vs. Non-Rigid Transformations 3

38 TASK B: Constructing Dilations (when center of dilation is outside a figure) 1. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the length CX. Transfer this distance along CX so that you find point X such that CX = 2(CX). Step 3: Repeat Step 2 with Points Y and Z. IMP Lesson: Rigid vs. Non-Rigid Transformations 4

39 Constructing Dilations (when center of dilation is on a figure) 2. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the length of CX. Transfer this distance along CX so that you find point X such that CX = 3(CX). Step 3: Repeat Step 2 with Points Y and Z. IMP Lesson: Rigid vs. Non-Rigid Transformations 5

40 Constructing Dilations (when center of dilation is inside a figure) 3. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the length of CX. Transfer this distance along CX so that you find point X such that CX = 2(CX). Step 3: Repeat Step 2 with Points Y and Z. IMP Lesson: Rigid vs. Non-Rigid Transformations 6

41 Objective: Students will be introduced to rotations, center of rotation, and angle of rotation. Students will make an angle template to use to rotate the same figure around a point on a figure, off the figure and within a figure. Rotation TASK A: Draw a quadrilateral in the first quadrant. Your quadrilateral should not intersect the x-axis, y- axis or the origin. Label the vertices A,B,C,D. Label the origin E. Follow the following steps to draw seven rotations of your pre-image. 1) Take one sheet of the patty paper and fold it in half and half again to make a smaller square and then in half on the diagonals. a. Trace the folds to make darker lines. This is the sheet students will record their rotations on. 2) Align the patty paper to the grid on the paper. Trace the figure and label the vertices 3) Turn the patty paper till the grid lines match, making sure E is in the same location. Trace the new figure and label the vertices. Repeat this process so that you will have the 7 rotations and the pre-image. Off the figure (Pre-image & 7 rotations) Word bank: Clockwise Counterclockwise Degrees Point Congruent Center of Rotation Interior Exterior Distance Figure Image Rotate Pre-Image 1) Describe the process you used to rotate the figure in the above activity. IMP Activity: Rotations 1

42 2) How did the size of the quadrilateral change each time it was rotated? How do you know? 3) Point E is called the center of rotation. Write a definition for center of rotation. 4) How did the distance from the origin to: all the images of Point A change? all the images of Point B change? all the images of Point C change? all the images of Point D change? How do you know? Point A: Point B: Point C: Point D: 5) If you rotated quadrilateral ABCD more times at a smaller interval, what would it look like if you connected all the images of D? 6) Write a precise definition of a circle: IMP Activity: Rotations 2

43 TASK B: Draw a quadrilateral where the center of rotation (origin) is on your figure (the origin should be a vertex of the quadrilateral or should fall on one of the sides of the quadrilateral). Label the vertices of your figure A,B,C,D. Follow the steps to draw 3 rotations of your pre-image. 1. Take one sheet of the patty paper and fold it in half and half again to make a smaller square. 2. Trace the folds to make darker lines. This is the sheet you will record their rotations on. On the figure (Pre-image & 3 rotations) 3. Align the patty paper to the grid on the paper. Trace the figure and label the vertices. 4. Turn the patty paper till the grid lines match, making sure the origin is in the same location. Trace the new figure and label the vertices. Repeat 2 more times. IMP Activity: Rotations 3

44 In the figure (Pre-image and 1 rotation) TASK C: Draw a quadrilateral where the center of rotation in inside your figure. Label the vertices of your figure A,B,C,D. Label the center E. Draw 1 rotation. 1. Take one sheet of the patty paper and fold it in half. 2. Trace the fold to make a darker line. This is the sheet you will record rotations on. 3. Align the patty paper to the grid on the paper. Trace the figure and label the vertices. 4. Turn the patty paper till the grid lines match, making sure the center of rotation is in the same location. Trace the new figure and label the vertices. IMP Activity: Rotations 4

45 TASK D: 7) The following figure ABC has been rotated. Find the angle of rotation. D is the center of rotation. Assume the figure was rotated counter-clockwise. a) Student 1: Draw CD and DC. Student 2: Draw BD and DB. Student 3: Draw AD and DA. b) Measure the angle created by the joined segments. Compare the measures with your neighbors. 8) What is the angle of rotation between the pre-image and your first image for each of the previous problems? a) TASK A: Justify. b) TASK B: Justify. c) TASK C: Justify. IMP Activity: Rotations 5

46 Practice: 1) In each of the following problems the figure ABC has been rotated. Find the angle of rotation. D is the center of rotation. Assume the figure was rotated counter-clockwise. a) b) Angle of rotation= Angle of rotation = c) d) Angle of rotation = Angle of rotation = Define Rotation: (Be sure to include points on the center of rotation and off the center of rotation in your definition) IMP Activity: Rotations 6

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51 Name: Date: Period: Perpendicular Bisectors TASK A: Define the perpendicular bisector of a segment: Sketch the perpendicular bisector of AB. Label the point of intersection C. Identify all the things you KNOW: A "Construction" in Geometry means to draw shapes, angles or lines accurately. Constructions use only compass, straightedge (i.e. ruler) and a pencil. TASK B: The steps for constructing the perpendicular bisector of a segment: B Construct the perpendicular bisector of AB. A 1 IMP Activity: Perpendicular Bisectors

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53 TASK C: Constructions: Perpendicular Bisectors In this section, we will discuss three different cases for constructing perpendicular lines. CONSTRUCTION #1: Construct the perpendicular bisector of a given segment. Given AB, construct the perpendicular bisector. 1. Choose any convenient setting that is more than 1 2 AB. 2. Set the compass at A and draw an arc through AB. 3. Keep the same setting, put the compass at B and draw an arc that intersects the previous arc both above and below the given segment. Label these points C and D. 4. Use a straightedge to draw CD. 5. CD is the perpendicular bisector of AB. A C D B CONSTRUCTION #2: Given a point on a line, construct a line through the point, perpendicular to the given line. Given point F on line l, construct a perpendicular at F. 1. Set the compass at F and draw two arcs on line l of M equal distance on either side of F. Label these H and K. 2. Using a LARGER setting for the compass, put the compass at H and draw and arc above the line. 3. Using the same setting, put the compass at K and draw an arc that intersects the previous arc. Label l this intersection point M. H F K 4. Use a straightedge to draw MF. 5. MF is perpendicular to line l at F. NOTE!! Construction #1 is applied when we need to find the midpoint of a segment.

54 CONSTRUCTION #3: Given a point not on a line, construct a line through this point perpendicular to the given line. Given line m and point (not Pon line m), construct a line P through P perpendicular to m. 1. Set compass at P and draw any arc that intersects line m in two points. Label these Q and S. 2. Set compass at Q, draw an arc below the line. 3. Using the same setting, put compass at S and draw an arc that intersects the previous arc. Label this T. Q 4. PT is perpendicular to line m. S m T NOTE!!: Construction #3 will help us when we reflect things over the line of symmetry. ASSIGNMENT: 1. Construct the perpendicular bisector of AB and CD. A C D B 2. Construct a line through G and H perpendicular to PQ and RS, respectively. P G Q R H S

55 3. Construct a line through J and K perpendicular to AB and CD. J K A B C D 4. Determine the midpoint of FL and HL. L F H 5. Construct the median (the segment connecting a vertex to the opposite midpoint) from R to side ST. R S T

56 6. For ABC, construct the three altitudes to each side. An altitude is a line segment through a vertex and perpendicular to the opposite side of the triangle. What appears to be true? A B C 7. Construct the bisectors for each side. Extend until they cross. Measure the distance from this point to each vertex. What seems to be true? D E F

57 Objective: Students will be able to construct a reflection of a figure and determine that the line of reflection is the perpendicular bisector of the line segments joining each pre-image and image point. Reflection by Construction TASK A: 1. Fold the patty paper along the diagonal. Darken in the crease with your pencil. Label the crease line l. 2. Draw ABC on the on one half of the of the patty paper. ABC should not intersect line l. 3. Fold the patty paper along line l and trace the image. Label the corresponding coordinates as A, B and C. 4. Draw AA, BB, CC. 1. Write at least three interesting observations that you have made. Be prepared to share with the class. 2. Record any observations your classmates made that you didn t. Staple your patty paper here: Complete the following considering what you observed: Reflection: a transformation in which a point B, not on the has an image, B, such that the is the of BB. IMP Lesson: Reflection by Construction 1

58 TASK B: Using the definition of a reflection and only an index card, reflect Trapezoid ABCD across the given line. Verify with a piece of patty paper that the image is congruent to the pre-image. 5. Record the order pairs for the image you graphed. 6. Describe the process you used to reflect the trapezoid. 7. When reflecting a point across a line: a) where will a point on the line end up? b) where will a point not on the line end up? IMP Lesson: Reflection by Construction 2

59 8. Based on what you have learned so far complete the statement. A line of reflection is also the of the segment connecting a pre-image and image because it the segment connecting each pre-image and image point and intersects each segment at a angle. Construct the reflection of point P across the given line. 1. Place your compass point on P and sweep an arc of any size that crosses the line twice. 2. Label points A and B. 3. Without changing the span on the compass, place the compass point at point A and make an arc on the opposite side of the line. Then place the compass point at point B and make an arc intersecting the previous arc. 4. Label the intersection P IMP Lesson: Reflection by Construction 3

60 TASK C: Using your compass and straightedge. 1. Reflect Point P across line m and label it P. 2. Reflect Point P across line n and label it P. 3. Reflect Point P across line m and label it P. 4. Reflect Point P across line n and label it P. Use the coordinate plane to justify that each of the points you found is the reflection it is supposed to be. Answer Key: P (4,2), P (-2,4), P (-4,-2), P (2,-4), P (4,2) IMP Lesson: Reflection by Construction 4

61 Objective: Students will explore the properties of isosceles triangles in relation to reflections. They will also practice constructing reflections and lines of reflections. Isosceles Triangles and Reflection Practice TASK A: 1. Reflect Triangle ABC across AB. 2. Identify all the congruent parts of the pre-image and image. 3. How do you know that all the parts of the pre-image and image are congruent? 4. What type of triangle is ACC? Justify your answer. 5. Do you think all isosceles triangles have a line of reflection? Why or why not? 6. Construct the line of reflection for isosceles triangle XYZ that maps Point X to Point Z. Label the intersection of segment XZ and the line of reflection Point M. 7. Identify all the congruent parts. 8. What can you conclude about point M? 9. What can you conclude about the angle created by the line of reflection and segment XZ? IMP Lesson: Isosceles Triangles and Reflections Practice 1

62 TASK B: On each of the coordinate grids there is a labelled point and a line. Let the line be a line of reflection. Find the image of the given point. Write the name of the image (e.g. B ) and its coordinates on the line provided (Hint: You will need a ruler and/or compass) IMP Lesson: Isosceles Triangles and Reflections Practice 2

63 For each given point and image, carefully find and draw the line of reflection. Write the coordinates of three points on the line of reflection. 1.,, 2.,, 3.,, 4.,, For at least two of the problems above, find the equation of the line of reflection. Find all four as a challenge. IMP Lesson: Isosceles Triangles and Reflections Practice 3

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65 -1- F l2j0k1a4h EKfu0t6a9 csho2futkwha9rwe1 JLjLQCR.g p 3Ail2l6 BrjizgShAtQsW FrZeJslewr2vNe4dS.P y XMBadd4eI KwJietRhl 5IQnjfSi1nKiotIep PGWe2o4mlegtNrFyP.R Worksheet by Kuta Software LLC Constructions Q.3 Construct a line segment congruent to each given line segment. 1) 2) Name Date Block Construct the perpendicular bisector of each. 3) 4) Construct a line segment perpendicular to the segment given through the point given. 5) 6)

66 -2- e A2J0B1Z47 1KkuStdaa msfozfwtmwgaer2eu 5LlLMCR.c q RAplblM qr Viog8hWtCsg WrOe7sleHrQvteTd3.d e 6MGaidae9 hwcigtihq liqntfsi7nqiytaep BGreyo4mke0tyr6yD.l Worksheet by Kuta Software LLC Construct a line perpendicular to the Construct a line parallel to the given line passing given line and passing through the given through the given point. 7) point. 8) Construct an equilateral triangle. Construct the line of reflection between the two points. 9) 10)

67 Name: Translation, Rotation, Reflection Tell how each figure was moved. Write translation, rotation, or reflection. Super Teacher Worksheets -

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69 COMMON CORE MATHEMATICS CURRICULUM IM1 Name Date Lesson: Reflections 1. Construct the line of reflection for the figures. 2. Reflect the given pre-image across the line of reflection provided. Lesson 14: Reflections 124 This work is derived from Eureka Math and licensed by Great Minds Great Minds. eureka-math.org This file derived from GEO-M1-TE This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

70 COMMON CORE MATHEMATICS CURRICULUM IM1 Sample Solutions 1. Construct the line of reflection for the figures. 2. Reflect the given pre-image across the line of reflection provided. Lesson 14: Reflections 125 This work is derived from Eureka Math and licensed by Great Minds Great Minds. eureka-math.org This file derived from GEO-M1-TE This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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73 Objective: Students will translate figures on the coordinate plane along a translation vector, describing their images and pre-image as parallel. Finally, students will go off the coordinate plane to construct translated figures. Translation Vectors Three types of transformations have been studied in this unit. The two rigid transformations we have already studied are rotations (turns) and reflections (flips). Today, we will study a third rigid transformation, translations. What word would you use to describe translation? A translation vector is an arrow giving a distance and a direction. The length of the vector tells how far to move each point and the arrow shows the direction. TASK A: Translate 1) each set of points according to the translation vector given. Label the points on the translated image with primes (A, B, etc.). Then answer the questions using the graphs. 1. a) Describe the translation in two ways, using words and coordinate notation. b) Look closely at ACand A C and boldly shade these two segments. What statements can you make comparing these two segments? Give reasons for your statements. c) What other pairs of segments will have the same relationship as ACand A C? IMP Activity: Translation Vectors 1

74 2. a) Describe the translation in two ways, using words and coordinate notation. b) Look closely at DD, EE, and FF and boldly shade these three segments. What statements can you make comparing these three segments? Give reasons for your statements. Word bank: angle, ordered pairs, translation vector, parallel, perpendicular, length, rigid motion, direction, congruent, and transformation Write a short paragraph about what translations are and how their images compare to their pre-images. Use at least 5 words from the word bank above. IMP Activity: Translations Vectors 2

75 TASK B: Using just a ruler, complete the two translations below following the given translation vectors as accurately as you can. Describe how you can check the accuracy of your translations. What tools can you use and how will you know if the translations have been done precisely? What tools other than a ruler might be useful in making accurate translations? List the tools and why they would be useful. IMP Activity: Translations Vectors 3

76 TASK C: Follow the directions on how to use a compass and straightedge to construct the following translation. 1. Draw a circle centered at A with radius DE. 2. Draw a circle centered at E with radius DA 3. Mark point A where the circles intersect 4. Draw a circle centered at B with radius DE 5. Draw a circle centered at E with radius DB 6. Mark point B where the circles intersect 7. Draw a circle centered at C with radius DE 8. Draw a circle centered at E with radius DC 9. Mark point C where the circles intersect 10. Draw A B C E D A Write a precise definition of translation. B C IMP Activity: Translations Vectors 4

77 Objective: Students will investigate and understand rotational symmetry in polygons. Students will find lines of symmetry, centers and angles of rotation and label the order of rotational symmetry for given polygons. Symmetries of Polygons A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation is said to have rotational symmetry. The number of times the figure lands on itself in a single full rotation is called the order of the rotational symmetry. TASK A: Use the diagram of the Ferris wheel to answer the following questions. a) As the ride makes one complete turn, in how many positions does the Ferris wheel look exactly the same as it does now? This is called the order of the rotational symmetry. Explain what you counted. b) What is the angle of rotation that first brings the Ferris wheel back onto itself? Explain how to find this angle without using a protractor. c) How many lines of symmetry does the Ferris wheel have? Ignore the hanging baskets and the legs that support the wheel. Justify your answer. d) What is the smallest and largest possible order of rotational symmetry for figures? Explain your reasoning. IMP Lesson: Symmetries of Polygons 1

78 TASK C: Complete the table with your group then answer the questions below. Lines of Symmetry(Y/N) # of Lines of Symmetry Rotational Symmetry(Y/N) Order of the rotational sym. Rectangle Angle of rotation Rhombus Parallelogram Trapezoid Equilateral Triangle Square Regular Pentagon Regular Hexagon You were given examples of regular polygons in this activity. Use the examples to write a definition for regular polygon. Regular Polygon: 1. Name all the regular polygons in the table above. Justify your answer. 2. What can you conclude about the number of lines of symmetry a regular polygon will have? 3. What can you conclude about the number of rotational symmetries a regular polygon will have? 4. Explain how to find the angle of rotation for the regular polygons. IMP Lesson: Symmetries of Polygons 2

79 TASK B: Student 1 1. A rectangle is a quadrilateral that has four right angles. Is it possible to reflect or rotate a rectangle onto itself? For the rectangle shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the rectangle onto itself. Describe the rotations and/or reflections that carry a rectangle onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) 2. A square is a quadrilateral that has four right angles and four congruent sides. Is it possible to reflect or rotate a square onto itself? For the square shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the square onto itself. Describe the rotations and/or reflections that carry a square onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) IMP Lesson: Symmetries of Polygons 3

80 TASK B: Student 2 1. A parallelogram is a quadrilateral where opposite sides are parallel. Is it possible to reflect or rotate a parallelogram onto itself? For the parallelogram shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the parallelogram onto itself. Describe the rotations and/or reflections that carry a parallelogram onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) 2. A regular pentagon is a 5 sided polygon with all sides congruent and all interior angles congruent. Is it possible to reflect or rotate a regular pentagon onto itself? For the regular pentagon shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the regular pentagon onto itself. Describe the rotations and/or reflections that carry a regular pentagon onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) IMP Lesson: Symmetries of Polygons 4

81 TASK B: Student 3 1. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Is it possible to reflect or rotate a trapezoid onto itself? For the trapezoid shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the trapezoid onto itself. Describe the rotations and/or reflections that carry a trapezoid onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) 2. A regular hexagon is a 6 sided polygon with all sides congruent and all interior angles congruent. Is it possible to reflect or rotate a regular hexagon onto itself? For the regular hexagon shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the regular hexagon onto itself. Describe the rotations and/or reflections that carry a regular hexagon onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) IMP Lesson: Symmetries of Polygons 5

82 TASK B: Student 4 1. A rhombus is a quadrilateral with four congruent sides. Is it possible to reflect or rotate a rhombus onto itself? For the rhombus shown below, find and label any lines of reflection, any centers of roation and angles of rotation that will carry the rhombus onto itself. Describe the rotations and/or reflections that carry a rhombus onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) 2. An equilateral triangle is a 3 sided polygon with all sides congruent. Is it possible to reflect or rotate a equilateral triangle onto itself? For the equilateral triangle shown below, find and label any lines of reflection, any centers of rotation and angles of rotation that will carry the equilateral triangle onto itself. Describe the rotations and/or reflections that carry an equilateral triangle onto itself. (Be as specific as possible in your descriptions. Use: diagonals, perpendicular, perpendicular bisectors ) IMP Lesson: Symmetries of Polygons 6

83 Name: Date: Period: Rigid Transformations Toolkit Key Vocabulary- Give a precise definition of each term Parallel lines: Perpendicular lines: Circle: Rigid transformation: Non-Rigid Transformation: Rotation: Reflection: Translation: Dilation: Construct the perpendicular bisector of AB Construct a line perpendicular to AB through P Construct a line perpendicular to AB through P A A A P B P B B IMP Toolkit: Rigid Transformations 1

84 Compare the characteristics of the three rigid transformations Characteristics of Translations Characteristics of Reflections Characteristics of Rotations Determine the symmetries of each figure Lines of Symmetry(Y/N) Number of Lines of Symmetry Rotational Symmetry(Y/N) Order of the rotational symmetry Angle of rotation Rectangle Rhombus Parallelogram Equilateral Triangle Square Regular Pentagon Regular Hexagon IMP Toolkit: Rigid Transformations 2

85 Mark the congruent and parallel parts on each transformation. Rotation: Find the coordinates of A and A Write the rule of rotation in coordinate notation: What is the center of rotation? What is the angle of rotation? Reflection: Draw the line of reflection. A Describe how you find the line of reflection: C 1 B 1 A 1 Translation: A Draw the translation vector. B Write the rule of translation in coordinate notation: C A 1 C 1 B 1 IMP Toolkit: Rigid Transformations 3

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87 IM 1 Unit 7 Transformation Review 1) Point X is the center of dilation. Construct dilations of ΔABC with scale factors of 2, 3, and 4. 2) Rotate square ABCD according to the following:

88 3) Rotate rectangle WXYZ according to the following: A a) 90 clockwise about A W X b) 90 counterclockwise about B c) 180 clockwise about Y 1 B Z Y 1 4) Rigid motions preserve and 5) Rigid motion is a sequence of in which points are moved by,, and/or but not by. 6) Construct the line of reflection for the figures in a, b, and c. a) b)

89 c) 7) Describe the series of transformations. a) b) c)