Geometric Transformations for Visualizing Shapes

1. Lesson 9-R Chapter 9 Review

2. Objectives • Review chapter 9 material

3. Vocabulary None new

4. Reflections - Flips Equal distance from line of reflection • Origin (x,y)  (-x, -y) Multiply both by -1 (origin is midpoint of all points and their primes) • Lines 1) x-axis (x,y)  (x, -y) Multiply y by -1(line y=0 acts as midpoint of all points and primes) 2) y-axis (x,y)  (-x, y) Multiply x by -1(line x=0 acts as midpoint of all points and primes) 3) line y = x (x,y)  (y, x) Switch x and y values(line y=x acts as midpoint of all points and primes) 4) horizontal line (y=k) similar in concept to x-axis, but no formula 5) vertical line (x=k) similar in concept to y-axis, but no formula

5. Translations - Slides • Transformation that moves all points of a figure, the same distance and direction • Translation function is the math effects or an equation relating old and new Don’t get fooled by order of appearance – focus on the words Down 3 and right 4 (x + 4 , y – 3) (x, y)  (x + 4, y - 3) Translation function

6. Rotations - Turns • A 180° rotation around the origin is the same as a reflection across the origin • 90° rotations around the origin can be done by measuring how far the point is from the closest axis. Use that distance to tell you how far away from the new axis the new point is • Remember the second grade method • Other rotations require trig to figure out changes based on rotational angle and point of rotation

7. Tessellation - Covering • Pattern using polygons that covers a plane so that there are no gaps or overlaps at a vertrex • Gaps occur if angles sum to less than 360° • Overlaps occur if angles sum to more than 360° • Regular Tessellation– formed by only one type of regular polygon. • Only regular polygons that tessellate are triangles, squares and hexagons. • Semi-regular Tessellation– formed by more than one regular polygons • Uniform – same figures (and angles) at each vertex

8. Dilations – Shrinks & Expansions • All dilations are similar figures • New point locations can be found graphically by drawing lines through endpoints and the center point and measure distance from center point • negative values for r mean the figure is on the opposite side of the center point • CT – congruence transformation

9. Misc Symmetry • Lines of symmetry allow you to fold a figure in half • A regular figure has the same number of lines of symmetry as it has sides • Rotational symmetry – a figure can be rotated less than 360° so that the pre-image and image look the same (indistinguishable) • Order – number of times figure can be rotated less than 360° in above (# of sides in a regular polygon) • Magnitude – angle of rotation (360° / order) • Point of Symmetry: midpoint between an point and its “folded” point • exists for regular, even sided polygons

10. y Vectors • Vector notation <x,y> vs Point notation (x,y) • Vector length – magnitude = √x² + y² • Vector direction – angle = tan (y/x) • Scalar multiplication: distribute constantk<x,y> = <kx,ky> • Vector addition: add components<a,b> + <c,d> = <a+c,b+d> Example: point (4,3) is the dot (white) vector <4,3> is the diagonal line (red) its x-component vector is the 4 part (yellow) its y-component vector is the 3 part (orange) Magnitude = √4² + 3² = 5 Direction: angle = tan (3/4) ≈ 37° x

11. Summary & Homework • Summary: • Translations, rotations and reflections are congruence transformations • Dilations are CT only for |r| = 1 • Lines of symmetry divided a figure in half • Tessellations are like tiles on the floor • Homework: • study for the test