TRANSLATION MATRIX

Definition Of Translation Matrix

A matrix representing a translated figure is a Translation matrix

Examples of Translation Matrix

• A'B'C'D' is the translation matrix of ABCD.

Video Examples: Matrix Translations

Solved Example on Translation Matrix

Ques: Express the translation of triangle ABC as the sum of a polygon matrix and a translation matrix.

Choices:

Correct Answer: A

Solution:

Step 1:Identify the coordinates of ABC and A'B'C'.
Step 2: The coordinates of ABC are A(- 3, 4), B(- 1, 2) and C(- 3, 2).
Step 3: The vertices of ABC in matrix form is 
                         A     B    C
                     
Step 4: The coordinates of A'B'C' are A'(1, - 2), B'(3, - 4) and C'(1, - 4).
Step 5: The vertices of A'B'C' in matrix form is
                         A'    B'    C'
                     
Step 6: Coordinates of image - Coordinates of pre image = Coordinates of the translation vector.
Step 7: The translation that maps ABC to A'B'C is 4 units left and 6 units down.
Step 8: Vertices of pre image + Translation matrix = Vertices of image.
Step 9:  -  = 

Ques: Find the coordinates of D and A' of the translation matrix of trapezoid. A table of the vertices of each trapezoid is shown below.

A. D(2, 2), A'(- 1, 5)
B. D(2, - 2), A'(- 1, 5)
C. D(- 2, - 2), A'(- 1, 5)
D. D(2, 2), A'(- 1, - 5)
Correct Answer: A 

Solution:

Step 1:Let (a, b) represent the coordinates of D and (c, d) represent the coordinates of A'.         
Step 2: Write the coordinates as a matrix equation. 
             
Step 3: 
Step 4:Solve an equation for x and y.
                     -2 + x = -1 Þ x = -1 + 2 Þ x = 1 
                     -4 + y = -3 Þ y = -3 + 4 Þ y = 1 
[Since, if two matrices are equal then their corresponding elements are equal.]
Step 5: Solve the equations for a, b, c, and d using the values x = 1 and y = 1.
                      a + x = 3 Þ a + 1 = 3 Þ a = 3 - 1 Þ a = 2 
                      b + y = 3 Þ b + 1 = 3 Þ b = 3 - 1 Þ b = 2 
                     -2 + x = c Þ -2 + 1 = c Þ c = -1 
                      4 + y = d Þ 4 + 1 = d Þ d  = 5 
Step 6: So, the coordinates of D(a, b) and A'(c, d) are D(2, 2) and A'(-1, 5).