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Proving Similarity via Transformations

Proving Similarity via Transformations. Dilation is a Non-Rigid Transformation that preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape. G-SRT Cluster Headings dealing with Similarity:

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Proving Similarity via Transformations

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  1. Proving Similarity via Transformations • Dilation is a Non-Rigid Transformation that preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape. • G-SRT Cluster Headings dealing with Similarity: • Understand Similarity in terms of similarity transformations • Prove theorems involving similarity

  2. Proving Similarity via Transformations • From a transformational perspective… • Two shapes are defined to be similar to each other if there is a sequence of rigid motions followed by a non-rigid dilation that carries one onto the other. • A dilation formalizes the idea of scale factor studied in Middle School.

  3. Prove Similarity by Transformations • What non-rigid transformation • proves that these triangles • are similar? • What is the center of dilation? • What is the scale factor of the • Dilation?

  4. Find Scale Factors Given a Transformation www.ck12.org Similarity Transformations Created by: JacelynO'Roark

  5. Circles in Analytic Geometry G-GPE (Expressing Geometric Properties with Equations) • Derive the equation of a circle given center (3,-2) and radius 6 using the Pythagorean Theorem • Complete the square to find the center and radius of a circle with equation x2 + y2 – 6x – 2y = 26 Think of the time spent in Algebra I on factoring Versus completing the square to solve quadratic Equations. What % of quadratics can be solved by factoring? What % of quadratics can be Solved by completing the square? Is completing the square using the area model more intuitive for students?

  6. Conic Sections – circles and Parabolas • Translate between the geometric description and the equation for a conic section • Derive the equation of a parabola given a focus and directrix • Parabola – Note: completing the square to find the vertex of a parabola is in the Functions Standards • (+) Ellipses and Hyperbolas in Honors or Year 4 Sketch and derive the equation for the parabola with Focus at (0,2) and directrix at y = -2 Find the vertex of the parabola with equation Y = x2 + 5x + 7

  7. Visualize relationships between 2-D and 3-D objects • Identify the shapes of 2-dimensional cross sections of 3-dimensional objects

  8. Visualize relationships between 2-D and 3-D objects • Identify 3-dimensional shapes generated by rotations of 2-dimensional objects http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html

  9. North country Inservice outline • Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences • Review of CCSSM Practice Standards – Share Experiences • Presentation of How Geometry Unfolds over K – 12 in CCSSM • Focus on Volume Standard in HS Geometry • Develop one unit focusing on HS Volume Standard and Practice Standards

  10. HS.GMD.A.1 • Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

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