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.

      

Solved Example on Translation Matrix

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

Choices:     

A.  +  =

B.  +  =

C.

D.  +  =

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:  -  =

One more example for Translation Matrix:

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

Choices:

A.

B.

C.

D.

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).

                               

Related Terms for Translation Matrix

• Slide
• Matrix

Additional Links for Translation Matrix

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