Unit 4 Guided Notes Part 2 Geometry

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1 Unit 4 Guided Notes Part 2 Geometry Name: Important Vocabulary: Transformation: A change in,, or of a geometric figure. Rigid transformation: A transformation that preserves measures and of segments. Transformation notation Preimage: The original figure (before it s transformed) Image: The figure that is a result of a transformation. Isometry: A transformation where the and are. Translation: Also known as a. It moves points the same distance and direction. Reflection: Also known as a. A transformation across a line of symmetry. Points on this line map to themselves. : Also known as a. All points rotation the same angle about the same point. Positive Direction = Counterclockwise. Translational symmetry: There is a translation that maps a figure to itself. Line/Reflectional symmetry: There is a that maps a figure to itself. al symmetry: There is a rotation (of less than ) that maps a figure to itself. ***We will hear more about the types of symmetry in a few lessons. Translations On Grid 1. Graph the triangle given by the coordinates A(-5, 2), B(-2, 3), and C(-3, 1). 2. a) Find the vertices for T<6,-4)(ABC) = A B C. A B C b) Graph A B C on the same grid as ABC. 3. A new triangle, A"B"C", is produced by transforming A B C using the rule: T<-2, 1>(A B C ) = A B C a) Using the coordinates from #2a as your original coordinates, find the coordinates for A"B"C". A B C b) Graph A"B"C" on the same grid as ABC and A"B"C". How are these three triangles related? c) Describe a rule you can use to obtain the image, A"B"C", by transforming the preimage ABC. State in words and as a symbolic rule.

2 In questions 1 3, you did two translations to produce the graph of triangle A B C. This is an example of a composition of transformations. A composition of transformations is a combination of two or more transformations. The composition of any two translations is another translation. 4. Find a translation that has the same effect as each composition of translations. a) T<5, 1>(x, y) followed by T<2, -6>(x, y) b) T<-2, 0>(x, y) followed by T<-4, 8>(x, y) 5. Graph the image of the figure for the translation 6. Write a rule to describe the translation. T <2,-3>(XYZ) Constructing Parallel Lines Practice Construct the line parallel to a given line AB through a given point P. P l A B Translations Off Grid Example 1 Use your compass and straightedge to apply T!" to segment P! P!. Note: Use the steps from the Parallel Line Construction twice for this question, creating two lines parallel to AB: one through P! and one through P!. P 1 A B P 2

3 Example 2 Use your compass and straightedge to apply T!" to P! P! P!. P 1 P 2 P 3 A B Reflections on Grid Instructions: For each coordinate grid given, plot the preimage using the given coordinate pairs and plot the image formed by performing the reflection. 1. A(4, 3) B(7, 1) C(6, -5) Reflect ABC across the y-axis 2. D(-5, 4) E(0, 7) F(5, -1) Reflect DEF across the x-axis What do you notice about the coordinates of A, B, and C compared to A, B, and C? What do you notice about the coordinates of D, E, and F compared to D, E, and F? 3. G(-6, 2) H(1, 4) I(4,0) Reflect GHI across the line y=-1 (Graph the line of reflection!)

4 Reflections Off Grid Example 1 Construct the segment that represents the line of reflection for quadrilateral ABCD and its image A B C D. What is true about each point on ABCD and its corresponding point on A B C D? Examples 2 3: Construct the line of reflection across which each image below was reflected Example 4: Construct the reflection of ABC over the line DE. Example 5 Now try a slightly more complex figure. Reflect ABCD across line EF.

5 s On Grid s about the origin 1. For each point, graph its rotation 90 o, 180 o and 270 o about the origin and put its coordinates in the table below. Original Location 90 o 180 o 270 o F(5, 1) F (-1, 5) F (, ) F (, ) A (, ) A (, ) A (, ) A (, ) L (, ) L (, ) L (, ) L (, ) C (, ) C (, ) C (, ) C (, ) O (, ) O (, ) O (, ) O (, ) N (, ) N (, ) N (, ) N (, ) 2. Using INDUCTIVE REASONING, what pattern do you notice about the coordinates from question 1? Write a thoughtful and complex observation using complete sentences, correct grammar and spelling s about a point not on the origin 3. For each point, graph its rotation 90 o, 180 o and 270 o about the point S and put its coordinates in the table below. Note: You do not need to graph any points that no longer Fall on this grid. Original Location 90 o 180 o 270 o F(5, 1) F (2, 4) F (, ) F (, ) A (, ) A (, ) A (, ) A (, ) L (, ) L (, ) L (, ) L (, ) C (, ) C (, ) C (, ) C (, ) O (, ) O (, ) O (, ) O (, ) N (, ) N (, ) N (, ) N (, ) 4. Using INDUCTIVE REASONING, what pattern do you notice about the coordinates from question 3? How does this compare to the pattern from question 1? Write a thoughtful observation. 5. Graph the rotation of the rectangle 90 o about the origin. 6. Graph the rotation of the rectangle 180 o about the point (- 1, 1) 7. Graph the rotation of the rectangle 270 o about the point (2, 0)

6 s Off Grid 1. Complete the following steps to draw the image of ΔXYZ under a rotation by m 1 in the positive direction about point T. a) Set your compass to the length of TZ. b) Place your compass spike on T and construct a large arc. It should intersect Z. This arc is like the orbit that your point will travel on as it is rotated. 1 c) Construct an arc that intersects both sides in 1 with your compass set to the length of TZ. It should look similar to the arc you drew in step b. Label these intersection points A and B. d) Measure the length of AB. Put your compass spike on Z and construct a small arc that intersects the big orbit arc. Z will be located at the intersection of your big arc and little arc. e) Do steps b & c for Y and then for X. (everywhere it says Z, replace with Y and X.) f) Construct XŹʹ, XÝʹ, and YŹʹ to complete ΔXʹYʹZʹ. T 2. Construct the image of ΔXYZ under a 90 (use the angle provided) rotation about T T

7 3. Construct the image of ΔXYZ under a rotation by m 2 in the positive direction about point T. T 4. Find the angle of rotation about D that maps the solid-line figure to the dashed-line figure. a) b) 5. Now that you can find the angle of rotation, let s move on to finding the center of rotation. Examine the figures below. Notice the points marked at the bottom left vertex and the bottom right vertex on the M and the image (dotted line version) of M. 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 A and A. b. Construct the perpendicular bisector of this segment. c. Construct a segment connecting B and B. d. Construct the perpendicular bisector of this segment. e. The point of intersection of the two perp. bisectors is the center of rotation. Label it P.

8 Exercises 6 7 Find the centers of rotation and angles of rotation for each pair of figures below. Exercise 6 Exercise 7 Types of Symmetry What types of symmetry does each logo below have? For reflectional, draw line(s) of symmetry. For rotational, give the smallest angle of rotation.