Take Note Properties: Rotation Matrices for the Coordinate Plane

Problem 3 Rotating a Figure

Rotate the triangle with vertices A(1, 1), B(5, 2), and C ( − 2 , 3 ) c open negative 2 comma 3 close  by the indicated amount. What are the vertices of the image? Graph the preimage and the image in the same coordinate plane.

1. 90 ° 90 to the degrees

   Plan

   How do you rotate the triangle?

   Multiply the 2 × 3 2 times 3  triangle matrix by the appropriate rotation matrix to get the 2 × 3 2 times 3  image matrix.

   • [ 0 − 1 1 0 ] [ 1 5 − 2 1 2 3 ] = [ − 1 − 2 − 3 1 5 − 2 ] table with 2 rows and 1 column , row1 column 1 , . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 1 , row2 column 1 , 1 , column 2 0 , end matrix . matrix with 2 rows and 3 columns , row1 column 1 , 1 , column 2 5 , column 3 negative 2 , row2 column 1 , 1 , column 2 2 , column 3 3 , end matrix , row2 column 1 , equals . matrix with 2 rows and 3 columns , row1 column 1 , negative 1 , column 2 negative 2 , column 3 negative 3 , row2 column 1 , 1 , column 2 5 , column 3 negative 2 , end matrix , end table

     The vertices of the image are ( − 1 , 1 ) , ( − 2 , 5 ) , open negative 1 comma 1 close comma open negative 2 comma 5 close comma  and ( − 3 , − 2 ) . open negative 3 comma negative 2 close .

     

2. 180 ° 180 to the degrees

   • [ − 1 0 0 − 1 ] [ 1 5 − 2 1 2 3 ] = [ − 1 − 5 2 − 1 − 2 − 3 ] table with 2 rows and 1 column , row1 column 1 , . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 0 , row2 column 1 , 0 , column 2 negative 1 , end matrix . matrix with 2 rows and 3 columns , row1 column 1 , 1 , column 2 5 , column 3 negative 2 , row2 column 1 , 1 , column 2 2 , column 3 3 , end matrix , row2 column 1 , equals . matrix with 2 rows and 3 columns , row1 column 1 , negative 1 , column 2 negative 5 , column 3 2 , row2 column 1 , negative 1 , column 2 negative 2 , column 3 negative 3 , end matrix , end table

     The vertices of the image are ( − 1 , − 1 ) , ( − 5 , − 2 ) , open negative 1 comma negative 1 close comma open negative 5 comma negative 2 close comma  and ( 2 , − 3 ) . open 2 comma negative 3 close .

     

✓ Got It?

3. Rotate the triangle with vertices D ( − 3 , 0 ) , E ( − 4 , 4 ) , d open negative 3 comma 0 close comma e open negative 4 comma 4 close comma  and F(1, 1) the indicated amount. What are the vertices of the image? Graph the preimage and the image in the same coordinate plane.
   1. 270 ° 270 to the degrees
   2. 360 ° 360 to the degrees