1. GBK Geometry Jordan Johnson

2. Today’s plan • Greeting • Lesson: Definitions of Transformations • Homework / Questions • Clean-up

3. Definitions of Transformations • Translation • Rotation • Reflection

4. Translation • A translation is a transformation which moves every point the same distance in the same direction. • Analytically: A translation by a and b maps every point P(x, y) in the pre-image to a corresponding point P(x + a, y + b) in the image (where a and b are constants).

5. Definition Strategy • Defining rotation and reflection of shapes: • Define rotation/reflection of points • Generalize that definition to shapes

6. Rotation • The rotation of a point A around a point P by  units produces a point A, such that PA = PA and APA = . • Unless  is a full- or half-revolution (180° or 360°), there are two such points, and we must indicate clockwise or counterclockwise rotation to determine which. Greek letter “theta”

7. Rotation • Generalized for rotating shapes: • A rotation rotates every point in the plane around the same center, in the same direction and by the same angle.

8. Reflection • The reflection of a point P across a line m is the point P such that m is the perpendicular bisector of PP. m

9. Reflection For reflecting shapes: • A reflection across a line m maps every point in the plane to its reflection across m.

10. Identity • The identity transformation is the transformation that maps every point to itself. • Examples: • Reflecting twice across the same line is equivalent to the identity transformation. • Rotating 360° is equiv. to the identity transformation.

11. More concise… • Some possible abbreviations: • “Reflectm” means “a reflection through line m.” • “RotateC, x” means “a rotation by x degrees around C.” • Id means “the identity transformation.” • T2○ T1 means “doing T2 after T1”(where T2 and T1 are transformations). • Conjecture: Reflectm○ Reflectm = Id • If m and n are lines that intersect at P, thenReflectm○ Reflectn = RotateP, 2θwhere θ is the size of an angle made by m and n.

12. Fixed Points • The fixed points of a transformation are the points that coincide with their own image.

13. Fixed points Transformation Fixed points • Translation by Δx, Δy • Rotation around P • Reflection across m • Identity • None • Point P • All points on line m • All points

14. Proof Problem • Take-home test • Due the day of the final • Start today • Don’t expect to solve it • Write out facts, things you need/want to know, things that would get you the solution • If you do solve it, look for a second way to solve it – there are at least four ways.

15. Assignments • Now: • Take out your chosen HW assignment from Nov/Dec. • Summarize: • When is it from? What assignment is it? • What makes it a good representative of your work? • What do you think you’ve improved upon, since then? • Put the assignment and summary in your folder. • Unit 4 Test w/Analysis – put it in your folder too. • Portfolio work • Find a construction to work on • Find a proof/problem to solve and write up • See me once you’ve found one

16. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!