Name Hr. Honors Geometry Lesson 9-1: Translate Figures and Use Vectors

Transcription

1 Name Hr Honors Geometry Lesson 9-1: Translate Figures and Use Vectors Learning Target: By the end of today s lesson we will be able to successfully use a vector to translate a figure. Isometry: An isometry is a that preserves length and angle measure. The three types of transformations are,, and. Translation Theorem: A translation is an isometry. Ex 1: Graph quadrilateral ABCD with vertices A( 2, 6), B(2, 4), C(2,1), and D( 2, 3). Find the image of each vertex after the translation (x, y) (x + 3, y 3). Then graph the image using prime notation. Rule: (x, y) (x + 3, y 3) A( 2, 6) A ( ) B(2, 4) B ( ) C(2, 1) C ( ) D( 2, 3) D ( ) Ex 2: Write a rule for the translation of ABC to A B C. Rule: (x,y) (, ) Vectors: A vector is a quantity that has both and. FG, read as vector FG. The Initial Point, or starting point, of the vector is. The Terminal Point, or ending point, of the vector is. The Component Form of a vector combines the horizontal and vertical components: < horizontal change, vertical change >..The component form of FG is <, > Ex 3: Name the vector and write its component form. a.) Name: Component form: b.) Name: Component form:

2 Ex 4: The vertices of ABC are A(0, 4), B(2, 3), and C(l, 0). Translate ABC using the vector < 4, 1>. Ex 5: A car heads out from point A toward point D. The car encounters construction at B, 8 miles east and 12 miles south of its starting point. The detour route leads the car to point C, as shown. a.) Write the component form of AB : <, > b.) Write the component form of BC : <, > c.) Write the component form of the vector that describes the straight line path from the car s current position C to its intended destination D *** Ex 6: The line y = 3x - 2 is translated using the vector 2, 1. What is the equation of the line after the translation? Ex 7: A translation of AB is described by CD. Find the value of each variable. CD = 1,3 A-12, 5c, A'1-3d,13 B3e + 2, 12, B'-6, 3f

3 Name Hr Honors Geometry Lesson 9-2: Use Properties of Matrices Vocabulary Description Illustration Matrix A matrix is a rectangular arrangement of numbers in rows and columns. Element Each number in a matrix is called an element. Dimensions The dimensions of a matrix are the numbers of rows and columns. Ex 1: a.) Write a matrix to represent the point or polygon. [ ] i.) Point A ii.) Quadrilateral ABCD b.) Write a matrix to represent ΔRST with vertices R( 5, 4),S( l,2),and T(3, 1). Ex 2: Add or Subtract the following Matrices. a.) = b.) = Ex 3: a.) The matrix represents ΔABC Find the image matrix that represents the translation of ΔABC 4 units left and 1 unit down. Then graph ΔABC and its image Translation matrix Polygon matrix Image matrix b.) The matrix represents ΔABC Find the image matrix that represents the translation of ΔABC 3 units right and 2 units up. Then graph ΔABC and its image. é ù é ê ú ëê ûú ê ê ë ù é ú ú = ê û ëê ù ú ûú Translation matrix Polygon matrix Image matrix:

4 Matrix Multiplication: In order to multiply two matrices, the # of in the first matrix must be equal the # of in the second matrix. If not, it is. The dimensions of the product are: ( of 1 st matrix) x ( of the 2 nd matrix). Ex 4: a) Do the columns of the 1 st = rows of the 2 nd? If so, what will the dimensions of the product be and find its its product? = ( x ) b) Do the columns of the 1 st = rows of the 2 nd? If so, what will the dimensions of the product be and find its its product? = ( x ) Ex 5: Multiply the following Matrices. a.) = b.) = *** Ex 5: Ex 6: A men s hockey team m needs 7 sticks, 30 pucks, and 4 helmets. A women s team w needs 5 sticks, 25 pucks, and 5 helmets. A hockey stick costs $30, a puck costs $4, and a helmet costs $50. Use matrix multiplication to find the total cost of equipment for each team.

5 Name Hr Honors Geometry 9-3: Perform Reflections Learning Target: By the end of today s lesson we will be able to successfully reflect a figure in any given line. Reflection Theorem: A reflection is an isometry. Ex 1: The vertices of Δ ABC are A(1, 2), B(3, 0), and C(5, 3). Graph the reflection of ΔABC described. a.) In the line : x = 2 b.) In the line m: y = 3 Ex 2: The endpoints of JK are J( l, 2) and K(l, 2). Reflect the segment in the given line. Graph the segment and its image. a.) In the line: y = x b.) In the line: y = x Coordinate Rules For Reflections Given point (a,b) If (a, b) is reflected in:.its image point is: x-axis y-axis y = x y = x Ex 3: a.) The vertices of Δ DEF are D (l, 2), E (2, 3), and F (4, 1). Find the reflection of Δ DEF in the y-axis. Graph Δ DEF and its image. Δ DEF D E F Pre-Image Image Matrix

6 Reflection Matrices: Reflection in the x-axis. Reflection in the y-axis Ex 4: The vertices of Δ QRS are Q ( l, 4), R (0, 1), and S (2, 3). Find the reflection of ΔQRS in the x-axis. What are the image points? Use matrix multiplication to verify the image. = Ex 5: Given the points A( 8,4), B( 1,3), find point C on the x-axis so AC + BC is a minimum. Step 1: Reflect one of the points over the x-asis. Step 2: Connect the image point with the other point. Step 3: Label the intesection C on x axis. *** This is the point which minimizes the sum of distance between two given points to a point on the x-axis. C(, ) *** Ex 6: The line y = 0.5x 4 is reflected in the line y = 2. What is the equation of the image?

7 Name Hr Honors Geometry Lesson 9-4: Perform Rotations Learning Target: By the end of today s lesson we will be able to successfully rotate figures about a point. Coordinate Rules For Rotations About The Origin: When a point (a, b) is rotated counterclockwise about the origin, the following are true: Rotation New Point Illustration (a, b) (, ) (a, b) (, ) Note: Rotations of a positive magnitude rotate. Rotations if a negative magnitude rotate. A rotation is an. Identity Matrix: Because a 360 rotation returns the figure to its position, the matrix that represents this rotation is called the identity matrix (a, b) (, ) (a, b) (, ) Ex 1: Graph quadrilateral KLMN with vertices K(3, 2), L(4, 2), M(4, 3), and N(2, 1). Then rotate the quadrilateral 270 about the origin. Rule for a rotation of 270, (a, b) (, ) K(3, 2) K (, ) M(4, 3) M (, ) L(4, 2) L (, ) N(2, 1) N (, ) Ex 2: Trapezoid DEFG has vertices D ( l, 3), E(1, 3), F{2,1), and G(1, 0). Find the image matrix for a 180 rotation of DEFG about the origin. Graph DEFG and its image. Rotation Matrix Polygon Matrix Image Matrix

8 Ex 3: Use quadrilateral DEFG in Ex 2. Find the image matrix after the rotation about the origin. a) 90 Rotation Matrix Polygon Matrix Image Matrix b) 270 Rotation Matrix Polygon Matrix Image Matrix c) 360 Rotation Matrix Polygon Matrix Image Matrix Rotation Theorem: A rotation is an isometry. Ex 4: The quadrilateral and triangle are rotated about P. Find the value of y and b. a.) b.) Ex 5: Rotate the following triangles about point P. a.) Draw a 150 rotation of ABC about P. Step 1: Draw a segment from A to P. Step 2: Draw a ray to form a 150 angle with PA Step 3: Draw A so that PA = PA. Step 4: Repeat Steps 1 3 for each vertex. Draw A B C. b.) Draw a 60 rotation of GHJ about P.

9 Honors Geometry 9-4 Supplement Notes 1. Reflection: x-axis 2. Rotation: 180 about the origin Rotation: 180 about ( 3, 0) Rotation: 90 about (0, 4) Honors Geometry 9-4 Supplement Notes 1. Reflection: x-axis 2. Rotation: 180 about the origin Rotation: 180 about ( 3, 0) Rotation: 90 about (0, 4)

10 Name Hr Honors Geometry Lesson 9-5: Apply Compositions of Transformations Learning Target: By the end of today s lesson we will be able to successfully perform combinations of two or more transformations. Composition of Transformations When two or more are combined to form a single transformation, the result is a composition of transformations. Composition Theorem: The composition of two (or more) isometries is an. Glide reflection A glide reflection is a combination of transformations in which every point P is mapped to a point P by the following steps: (1) A. And (2) A. Ex 1: The vertices of ΔABC are A(2,1), B(5, 3), and C(6, 2). Find the image of ΔABC after the glide reflection. Translation: (x, y) (x 8, y) Reflection: in the x-axis Ex 2: The endpoints of CD are C( 2, 6) and D( l, 3). Graph the image of C"D" after the composition. a.) Reflection: in the y-axis, followed by a Rotation: 90 about the origin. b.) Rotation 90 about the origin, followed by a Reflection in the y-axis

11 Explore: Graph the parallel lines x= -2 and x= 4. Graph the given segment with endpoints C( 6, 4) and D( 4, -3). Reflect the segment over x = -2, then over x = 4. Observations: 1. Compare CD with C "D". What type of transformation results from reflecting the segment over two parallel lines? 2. Compare the distance between the two parallel lines and the distance between C and C and D and D? What relationship do you see? Reflections In Parallel Lines Theorem: If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a. If P is the image of P, then: 1) PP =, where d is the distance between k and m. 2) PP is to k and m. Ex 3: In the diagram, a reflection in line k maps GF to G ' F'. A reflection in line m maps G ' F' to G "F". Also, FA = 6 and DF = 3. a.) Name any segments congruent to each segment: GF, FA, and GB. b.) Does AD = BC? Explain. c.) What is the length of GG "? Reflections In Intersecting Lines Theorem: If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a about. The angle of rotation is, where x is the measure of the or right angle formed by k and m. Ex 4: In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps F to F and A to A.

12 Name Hr Honors Geometry Lesson 9-6: Identify Symmetry Learning Target: By the end of today s lesson we will be able to successfully identify line and rotational symmetries of a figure. Ex 1: How many lines of symmetry does each figure have? a.) b.) c.) d.) e.) ***Notice that the lines of symmetry are also lines of. Rotational Symmetry: A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a of 180 or less about the of the figure. For a figure with s symmetry lines, the smallest degree of rotation that maps the figure onto itself is: Ex 2: Figure Draw ALL Symmetry Lines # of Symmetry Lines Smallest Degree of Rotational Symmetry? Rotational Symmetry? Y or N? If yes, what are other degrees of rotation? Square Regular

13 Figure Draw ALL Symmetry Lines # of Symmetry Lines Smallest Degree of Rotational Symmetry? Rotational Symmetry? Y or N? If yes, what are other degrees of rotation? Ex 3: Does the figure have the rotational symmetry shown? a) 135 b) 180 If not, does the figure have any rotational symmetry? *** Ex 4: Identify all lines of symmetry and all angles of rotation that map the figure onto itself. I. II. a) Describe a way to shade exactly three parallelograms in the figure and still have the same line symmetry and rotational symmetry. b) Describe a way to shade three parallelograms in the figure so that there is one line of symmetry and no rotational symmetry. c) Describe the least number of triangles that can be shaded in the figure so that there is exactly one line of symmetry. Does the rotational symmetry change? d) Describe the triangles that could be shaded in the figure so that the figure has the greatest number of line and rotational symmetries possible.

14 Name Hr Honors Geometry Lesson 9-7: Identify and Perform Dilations Learning Target: By the end of today s lesson we will be able to successfully use drawing tools and matrices to draw dilations. Ex 1: Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. a.) b.) CP' =, CP CP' CP =, Scale Factor, k =. The image P is a(n). Scale Factor, k =. The image P is a(n). c.) Find the value of x. P'R' PR =, Scale Factor, k =. The image P is a(n). Find x: x =. Ex 2: Simplify the product: a.) b.) c.) Ex 3: The vertices of quadrilateral ABCD are A ( 3, 0), B (0, 6), C (3, 6), and D (3, 3). Use scalar multiplication to find the image of ABCD after a dilation with its center at the origin and a scale factor of ⅓. Graph ABCD and its image. Ex 4: The vertices of ΔRST are R ( 4, 3), S ( l, 2), and T (2, 1). Use scalar multiplication to find the vertices of ΔR S T after a dilation with its center at the origin and a scale factor of 2.

15 Ex 5: The vertices of ΔKLM are K( 3, 0), L( 2,1), and M( l, 1). Find the image of ΔKLM after the given composition. Translation: (x, y) (x + 4, y + 2) Dilation: centered at the origin with a scale factor of 2 Ex 6: A segment has the endpoints C ( 2, 5) and D (3, -1). Find the image of CD after a 90 rotation about the origin followed by a dilation with its center at the origin and a scale factor of 3. Ex 7: a) Construct a dilation of parallelogram LMNP with b) Construct a dilation of parallelogram LMNP with point L as the center of dilation and a scale factor of ½. point X as a center of dilation and a scale factor of ½. Ex 8: a) Construct a dilation of triangle PQR with point b) Construct a dilation of triangle PQR with point X as the P as the center of dilation and a scale factor of 2. as the center of dilation and a scale factor of 2.