1. Honors Geometry

2.  We learned how to set up a polygon / vertex matrix  We learned how to add matrices  We learned how to multiply matrices

3.  I will learn how to use matrices to model transformations.

4.  We use matrix addition to translate a figure  We must set up the two matrices ourselves  Polygon matrix  Translation matrix  Review rule for matrix addition

5.  This matrix will be added to the polygon matrix, so they must be the same dimension  Once dimension is set, the top row of the matrix will reflect the horizontal shift of the translation, vertical on the bottom  if left, negative, if right, positive  This is the information from the translation vector

6.  Set up a translation matrix that will shift a triangle 4 units right and 2 units down  Triangle: 2 x 3  Translation matrix:

7.  Shift the triangle ABC with vertices A(2, 4) B(4, 1) C(3, -3) along the vector  Polygon matrix:  Translation matrix:  Add! A’ B’ C’

8.  Quadrilateral EFGH with vertices E(-3, 2), F(-2, 4) G(4, 1) H(3, 0) is translated 1 unit left and 3 units down.  Write a translation matrix  Find the coordinates of E’F’G’H’ E’(-4, -1) F’(-3, 1) G’(3, -2) H(2, -3)  Graph the image and the preimage

9.  Graph both

10.  Both rotations and reflections use matrix multiplication  We use a special set of 2 x 2 matrices to perform specific rotations and reflections  Review rule for matrix multiplication  Why will it always be possible to multiply a polygon matrix by a 2 x 2 special matrix?

11.  Reflection (based on line of reflection)  Rotation (based on angle of rotation) x axis origin y axis y = x 90° 180° 270° 360°

12.  Notice: reflection about the origin and 180 degree rotation have the same special matrix  Why?  What does a 360 degree rotation do? Why is its matrix “extra” special?

13.  When multiplying by one of our 8 special matrices, always put the special matrix on the left (first)  Remember that matrix multiplication is NOT commutative  AB ≠ BA

14.  Reflect triangle ABC with vertices A(-3, 1) B(1, 3) C(2, 0) across the x axis  Our two matrices:  Multiply! A’ B’ C’

15.  Triangle XYZ with vertices X(1, -3) Y(-4, 1) Z(-2, 5) is rotated 180 degrees about the origin  Find the coordinates of X’Y’Z’ X’(-1, 3) Y’(4, -1) Z’(2, -5)  Graph the image and the preimage

16.  Graph