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Geometric Transformations and Algebraic Connections

Explore transformations in the plane, from translations to rotations and reflections. Understand dilations and their properties. Connect algebra with geometry through proving theorems using coordinates and solving geometric problems algebraically. Learn about circle centers, parallelograms, rectangles, squares, rhombi, and more. Practice with examples and review key concepts.

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Geometric Transformations and Algebraic Connections

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  1. Unit 5 – Transformations in the Plane Unit 6 – Connecting Algebra with Geometry EOCT Review

  2. Key Ideas • Precise definitions: • Angle • Circle • Perpendicular lines • Parallel lines • Line Segment

  3. Unit 5 - Transformations • Represent transformations in the plane • Compare rigid and non-rigid • Translations • Rotations • Reflections • Understand Dilations

  4. Key Ideas • Given shapes – • Determine which sequence of rotations and reflections would map it on itself • Develop definitions of rotations, reflections and translations

  5. Translations A translation maps every two points P and Q to points P' and Q' so that the following properties are true: ● PP' = QQ' ● PP' QQ'

  6. Reflections A reflection across a line m maps every point R to R' so that the following properties are true: ● If R is not on m, then m is the perpendicular bisector of RR' . ● If R is on m, then R and R' are the same point.

  7. Rotations A rotation of x° about a point Q maps every point S to S' so that the following properties are true: ● SQ = S' Q and m<SQS' = x° . ● Preimage point Q and image point Q' are the same. Note: QS and QS' are radii of a circle with center Q.

  8. Examples Use the translation (x, y) → (x – 3, y + 1).

  9. Examples Describe every transformation that maps the given figure to itself.

  10. Unit 6 – Connecting Algebra and Geometry • Use coordinates to prove simple geometric theorems algebraically. • Circle centers • Parallelograms • Rectangles • Squares and Rhombii

  11. Geometric Problems • Prove lines parallel or perpendicular • The slope of the line through points (x1, y1) and (x2 , y2 ) is • Find the equation of a line through a given point • y=mx+b

  12. Distance Formula

  13. Exterior Angle Theorem m<1= m<2+m<3

  14. Key Ideas • Partition a directed segment into a given ratio.

  15. Review Examples • See Study Guide Handout

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