Isometry: When the preimage and image are congruent. It is a motion that preserves the size and shape of the image as it is transformed.

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1 Chapter Notes Notes #36: Translations and Smmetr (Sections.1,.) Transformation: A transformation of a geometric figure is a change in its position, shape or size. Preimage: The original figure. Image: The final figure after a transformation has occurred. Isometr: When the preimage and image are congruent. It is a motion that preserves the size and shape of the image as it is transformed. Translations: A translation is a transformation of points on a graph in which a set of points slides, or shifts, location. All points affected b a single translation, or slide, must shift the same distance in the same direction. 1.) We are going to translate Δ ABC 3 units to the right and units down b moving points A, B, and C in this manner to points A, B, and C, respectivel. Write the coordinates of each point: A A B B C C We describe this translation as (this is the rule for the translation): (, ) (, ) ** Δ ABC is called the of this translation and Δ ABC ' ' ' is called the of this translation. **.) Find the image of the figure under the given translation. (, ) ( 6, + 3)

2 Complete: 3.) (, ) (, + 1) a) This translation glides points units left and units up b) The image of (, 6) is (, ) c) The preimage of (, 6) is (, ).) (, ) ( +, - ) a) This translation glides points and b) The image of (, 6) is (, ) c) The preimage of (, 6) is (, ) State whether the transformation appears to be an isometr..) 6.) pre-image pre-image image image Smmetr: There are three tpes of smmetr: reflectional/line, rotational, and point. Reflectional Smmetr/Line Smmetr How man lines can ou draw through the heagon that make mirror-image congruent halves? This is an eample of reflectional or line smmetr. One half of the figure is a mirror image of its other half. Rotational Smmetr a) b) A figure has rotational smmetr if its own image is created b rotating the image 10 0 or less. Which of these triangles have rotational smmetr? How man degrees do ou need to rotate it to get a smmetric figure?

3 Point Smmetr a) b) A figure has point smmetr if it can be rotated about a point 10 0 and creates the same image. This is a special tpe of rotational smmetr. Which figures have point smmetr? c) d) Tell what tpe(s) of smmetr each figure has. If it has line smmetr, sketch the line(s) of smmetr. If it has rotational smmetr, state the angle of rotation. 7.).).).) Notes 37: Reflections, Rotations Reflections: A reflection is a transformation of points on a graph in which a line acts like a mirror sending points images to new locations on the graph. Pretend that line m is a mirror. Reflect points A, B, and C about line m. Name their images A, B, and C, respectivel. This reflection in line m is called: We describe individual mappings as: Reflection of A across m maps A to A, or: A A Write the two other reflections shown:

4 Complete the following. Note the reflections in lines h and k. A j D H B E G I 7.) Reflection across j stands for C.) Refection across h: HG F )Reflection across h: A 11)Reflection across j: DEF J ) Reflection across j : A 1) Reflection across h: AB h Reflect each image in line p: 13.) 1.) 1.) Write the coordinates of the image of each point b reflection in (a) the -ais, (b) the -ais, and (c) the line =. 16.) A a) the -ais b) the -ais c) = 17.) B a) the -ais b) the -ais c) =

5 Given points T(, ), A(-3, -), B(0, -), draw TAB and its reflection image across each line. 1.) = -3 1.) = 0.) = + Rotations: A rotation is another eample of a transformation in which points images are sent to new locations on our graph. You will need to imagine sticking our original point to a steering wheel and then rotating the wheel a given number of degrees. Where the point stops is its image under a rotation. Rotations are described based on the point of rotation (the center of our steering wheel) and the angle of rotation (how man degrees ou turn our wheel). Counterclockwise rotations are indicated b positive degree measures. Clockwise rotations are indicated b negative degree measures.

6 Check out these rotations. (I mapped P to P and called the center of the wheel H) 0 0 rotation about H -0 0 rotation about H 00 0 rotation about H Ke degree measurements to remember: : a quarter of the wa around a circle : half wa around a circle : all the wa around a circle Name each image point: eamples: 0 0 Rotation of L about O. Start at point L, rotate the wheel 0 degrees counterclockwise (0 is positive) around point O, and name the point where ou would end up:.) 0 0 rotation of N about O 6.) Rotation of M about O. 7.) 0 0 Rotation of P about O. N M P O L Q R State another name for each rotation: (hint: switch direction of rotation and find a new wa to get to that spot on the wheel).) 0 Rotation of P about O.) -0 0 Rotation about O.) -13 Rotation about O O

7 Notes # 3: Dilations and Tessellations (Sections.,.7) Dilations: A dilation is a transformation in which make the pre-image proportionall larger (an enlargement) or smaller (a reduction) b a given scale factor. The image will alwas be similar to the pre-image. We are going to dilate Δ ABC and shrink it to half its size. This would be a reduction. One such dilation is: Dilation with center O and scale factor of ½ : ΔABC Δ ABC ' ' ' This means: Graph points A, B, and C so that OA = ½ OA, OB = ½ OB, OC = ½ OC Steps: - measure OA and mark its midpoint as A - measure OB and mark its midpoint as B - measure OC and mark its midpoint as C - connect A, B, and C as a triangle - ΔABC ' ' ' is similar to and half the size of Δ ABC ) Sketch the image under the given dilation: Dilation with a center of O and a scale factor of. You need to construct a triangle that is similar to ΔXYZ with a scale factor of. Meaning, our new points X, Y, and Z must be twice the distance than the given points from O. This is an eample of an enlargement. X Y O Z

8 1.) Find the coordinates of the images of A, B, and C b the given dilation. Dilation with center O and scale factor of A (-, ) A (, ) B (0, -3) B (, ) C (, 1) C (, ) The blue figure is a dilation image of the green figure. Describe the dilation. 13.) 1.) 1.)

9 Tessellations A tessellation is a repeated pattern of figures that completel covers a plane without gaps or overlas. The sum of the measures around an verte must be All triangles tessellate 10( n ) a = for triangles n = 3 n 10(1) a = = 60 and 60 is a factor of All quadrilaterals tessellate n = 10() a = = 0 and 0 is a factor of 360. Determine whether each figure will tessellate a plane..) rhombus.) acute triangle 6.) regular heagon 7.) regular decagon.) rectangle

10 Reflections: 1. Complete each statement: A j D Geometr Chapter Stud Guide. Reflect the image in line p 3. Write the coordinates of point A b reflection in the (a) -ais, (b) -ais, and (c) the line = B E h C F a) Reflect across h: BC b) Reflect across j: BC c) Reflect across j : DEF Translations:. Complete: (, ) ( -, + 3) a) Translation glides points (a) (b) (c). Point P and its image P are shown. Complete the translation statement. (, ) (, ) b) The image of (3, -) is (, ) c) The preimage of (3, -) is (, ) Rotations: 6. State another name for the rotation: (a) 0 rotation about O 7. Name each image point: (a) 0 Rotation of Q about O N M (b) rotation about O (b) 0 Rotation of M about O P O L Dilations: Complete each dilation. Center at O; Scale factor ½ : ΔABC Δ A' BC ' ' Q R. Center at O; Scale factor of - : ΔABC Δ A' BC ' '

11 Indicate how man lines of smmetr and what tpe of smmetr for each figure: H.) 11.) 1.) 13.) 1.) Which of these figures will tessellate? 17) 1)

12 HW#0: Final Eam Review 1.) The dimensions of a rectangle are m b m. Find the length of its diagonal..) The perimeter of a square is 0m. Find its area. Name: 3.) The hpotenuse of a --0 triangle is 1cm. Find the length of its legs..) The shortest leg of a triangle is in. Find the length of the other two sides of the triangle..) The longer leg of a triangle is 1m. Find the length of the other two sides of the triangle. 6.) The perimeter of a regular heagon is 60cm. Find its area. 7.) Write a trig equation to solve for..) Write a trig equation to solve for..) Write a trig equation to solve for. 1.) The circumference of a circle is 16π in. Find the area of the circle. 11.) The perimeter of an equilateral triangle is 7m. Find its area. 3 1.) The diagonals of a rhombus are m and 1m. Find its area. 13.) Find the lateral area and volume of a clinder with height 3in and radius in. (LA = π rh, V = π rh) 1.) Find the surface area of a sphere with diameter 1m. (SA = π r ) 1.) Find the lateral area of a square pramid with base edge 6in and height in. (LA = Pl )

13 16.) Solve for : 0 17.) PQ = 16, OM =, Diameter = P O M Q 1.) CD=1, OM=, ON=, EF =, radius =, diameter = D M O E N C F 1.) =, = 60 0.) =, = 1.) =, =, z = z 3.) = 60 3.) = 30.) = ) = 6.) = 7.) Find the center and the radius of the circle: ( + 1) + ( ) = 6

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