1. 7.4 Translations and Vectors June 23, 2016

2. Goals Identify and use translations in the plane. Use vectors in real- life situations.

3. Translation A transformation that maps every two points P and Q in the plane to points P’ and Q’, so that: –PP’ = QQ’ –PP’ // QQ’, or PP’ and QQ’ are collinear P P’ Q Q’

4. Translation Theorem (7.4) A translation is an isometry.

5. Finding the image of a Translation By gliding a figure in the plane. Complete one reflection after another in two parallel lines. Thm. 7.5

6. Theorem 7.5 If lines k and m are parallel, then a reflection in line k followed by a reflection in line m, is a translation. If P’’ is the image of P, then the following is true: –PP’’  k and PP’’  m –PP’’ = 2d (d is the distance between k and m) Q P P’ Q’ P” Q” k m 2d d

7. Translations in a Coordinate Plane (x,y)  (x + a, y + b) Each point shifts a units horizontally and b units vertically. Ex. (x, y)  (x + 4, y – 2) shifts each point 4 units to the right and 2 units down.

8. More Vocabulary Vector: another way to describe a translation; a quantity that has direction and magnitude (size) and is represented by an arrow drawn between 2 points.

9. More Vocabulary Initial point: the starting point of the vector (P). Terminal point: the ending point (Q) Component form: combines the horizontal (5) and vertical (3) components of vector PQ; P Q 5 units right 3 units up

10. Example 1 Sketch a parallelogram with vertices R(-4, -1), S(-2, 0), T(-1, 3), U(-3, 2). Then sketch the image of the parallelogram after translation (x, y)  (x + 4, y – 2).

11. Example 2 In the diagram, name each vector and write its component form. C D A B

12. Example 3 The component form of vector RS is [2, -3]. Use vector RS to translate the quadrilateral whose vertices are G(-3, 5), H(0, 3), J(1, 3), K(3, -2).

13. Example 4 The initial point of a vector is V(-2, 3) and the terminal point is W(-4, -7). Name the vector and write its component parts.

14. Example 5 AB  A’B’ using a translation. The coordinates of the endpoints of AB are A(-2, 1) and B(3, -1). The coordinates of A’B’ are A’(1, -2) and B’(6, -4). Write the component form of the vector that can be used to describe the translation.

15. Example 6 What is another name for a translation? What is another name for rotation? What is another name for reflection?

16. Homework Pg 425 #’s 2-34 even, 39, 40