Warm Up 1. Reflect the preimage using y=x as the line of reflection given the following coordinates: A(-2, 4), B(-4, -2), C(-5, 6) 2. Rotate the figure 90 degrees counterclockwise given the following coordinates. D(0, 4), E(3, -4), F(2, 2)

Translations Using Vectors and Matrices Chapter 7.4

Translations Translation Theorem A translation is an isometry. 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: 1.PP” is perpendicular to k and m. 2.PP” = 2d, where d is the distance between k and m. k m Q P Q’ P’ P’’ Q’’

Example 1: Translate the following figure using the given coordinate. Coordinate Notation: (x,y) →(x-3,y+4)

Transformations using Vectors Vector ⇀ : a quantity that has both direction and magnitude (length) and is represented by an arrow drawn between two points. Initial point: starting point (P) Terminal point: ending point (Q) Component form:x 2 -x 1,y 2 -y 1 Magnitude: distance between A and B (distance formula) P(x 1,y 1 ) Q(x 2,y 2)

What can we state about this vector? The vector is named PQ The magnitude (length) of the vector is Component form: 7-2,8-2 5,6 P(2,2) Q(7,8) ⇀ ⇀

Example 2: Find the image of D(1, -2) under the translation described by the vector. Find the magnitude of the vector y x D(1,-2); D’(2,2) ⇀

Warm Up Add the following matrices: 1. + = 2. + =

Example 3: Use a matrix to find the image of each figure under the given translation.

Transformations Using Matrices Use the matrices to find the image of under the translation. M F H X-coordinate Y-coordinate Translation Matrix: Add the matrices: + =