TRANSFORMATIONS WITH MATRICES
Section 44
Points on the coordinate plane can be represented by matrices. The ordered pair (x, y) can be represented by the column matrix at the right.
Polygons on the coordinate plane can be represented by placing all of the column matrices into one matrix called a vertex matrix.
ΔABC with vertices A(3, 2), B(4, 2 ), and C(2, 1) can be represented by the following vertex matrix. A
C B You can use matrices to perform transformations. (translations, reflections, and rotations)
Remember that the original figure is called the preimage and the figure after the transformation is the image.
If the two figures are congruent then the transformation is an isometry.
Example 1 Translation a. Find the coordinates of the vertices of the image of quadrilateral QUAD with Q(2, 3), U(5, 2), A(4, 2), and D(1, 1), if it is moved 4 units to the left and 2 units up.
Write the vertex matrix for quadrilateral QUAD.
Write the transformation matrix. Example 1 continued
Vertex Matrix Translation Vertex Matrix of QUAD Matrix of Q'U'A'D'
+ =
The coordinates of Q'U'A'D' are: Q'(2, 5), U'(1, 4), A'(0, 0) and D'(3, 1).
b. Graph the preimage and the image.
Example 2 Rectangle A'B'C'D' is the result of a translation of rectangle ABCD. A table of the vertices of each rectangle is shown. Find the coordinates of A and D'.
Rectangle Rectangle ABCD A'B'C'D' A A'(1, 1) B(1, 5) B'(4, 1) C(1, 2) C'(4, 6) D(4, 2) D' Dilations When a figure is reduced or enlarged it is called a dilation. All linear dimensions of the preimage change in the same ratio.
Example: If the length of each side of a figure doubles, then the perimeter doubles, and vice versa.
When a dilation occurs, the figures are not congruent, they are similar.
Therefore, Dilations are not isometries.
You can use scalar multiplication to perform dilations.
Example 3 Dilation
ΔJKL has vertices J(2, 3), K(5, 4), and L(3, 2). Dilate ΔJKL so that its perimeter is onehalf the original perimeter.
a. Find the vertices of ΔJ'K'L'.
Multiply the vertex matrix for ΔJKL by the scale factor ½ to find the vertices of ΔJ'K'L'. Example 3 continued
The coordinates of the vertices of ΔJ'K'L' are J'(1, 1½), K'(2½, 2), and L'(1½, 1).
b. Graph the preimage and the image.
Reflections A reflection occurs when every point on a preimage is reflected across a line of symmetry using a reflection matrix. Reflection Matrices
For a reflection over the: xaxis yaxis line y = x
Multiply the vertex matrix on the left by: Example 4 Reflection
Find the coordinates of the image of pentagon PENTA with P(4 ,3), E(1, 4), N(1, 3), T(0, 1), and A(3, 1) after a reflection across the line y = x.
Write the vertex matrix and multiply it by the reflection matrix for the line y =x.
x =
Example 4 continued The coordinates of PENTA are: P(4 ,3), E(1, 4), N(1, 3), T(0, 1), and A(3, 1). The coordinates of P'E'N'T'A' are: P'(3, 4), E'(4, 1), N'(3, 1), T'(1, 0), and A'(1, 3).
Graph the line y = x, PENTA, and P'E'N'T'A'. Rotations A rotation occurs when a figure is moved around a center point, usually the origin (0, 0). To determine a figure's image coordinates, multiply its vertex matrix by a rotation matrix.
Rotation Matrices For a counterclockwise rotation about 900 1800 2700 the origin of:
Multiply the vertex matrix on the left by:
Example 5 Rotation
Find the coordinates of the image of ΔABC with A(4, 3), B(2, 1), and C(1, 5) after it is rotated 900 counterclockwise about the origin.
Write the vertex matrix and multiply it by the rotation matrix.
x = Example 5 continued The coordinates of ΔABC are: A(4, 3), B(2, 1), and C(1, 5). The coordinates of ΔA'B'C' are: A'(3, 4), B'(1, 2), and C'(5, 1).
Graph ΔABC and ΔA'B'C'