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Mathematical Investigations II
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Mathematical Investigations II
MATRICES & GEOMETRIC TRANSFORMATIONS
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There is a simple mathematical way of describing a polygon that has been drawn on coordinate graph
paper. We can "encode" its vertices in a rectangular array of numbers called a matrix. The array we use
will have two rows and as many columns as the polygon has vertices. We will write the ordered pair of
each vertex as one column of the matrix. The array will be enclosed with left and right square brackets.
So the matrix for ABC with vertices A(–8, 7), B(–4, 10) and C(–1, –3) from Matrices 2.2 is as follows:
A
B
C
x  8 4 1
y  7 10 3
We will call this the vertex matrix or coordinate matrix, for ABC . The dimensions of this matrix are
said to be 2  3 (read “2 by 3”), which means that the matrix has two rows and three columns. The
number of rows is always given first.
Two matrices (the plural of matrix) can be added only if they have the same number of rows and
columns (in other words, the same dimensions). For instance, the matrices below are added as shown:
 8 4 1  1 2 4  8  1 4  2 1  4  7 6 3 
 7 10 3   3 5 2   7  3 10  5 3  2    4 15 1

 
 
 

Reflect on what you did on exercises 1, 2, 3, 5, 6, and 8 of worksheet Matrices 2. The vertex matrix
(pre-image) for the triangle is given. Fill in the matrix for its image under each transformation. Then try
to write a translation matrix that you can add to the pre-image triangle to get the image triangle.
Pre-image
A
1.
B
Image
A
C
x  8 4 1 x 

y  7 10 3 y 
A
2.
B
Translation
C
 x
 y 





A
C
x  8 4 1 x 

y  7 10 3 y 
B
 x
 y 


Matrices. 3.1
B
C 



Rev S11
Mathematical Investigations II
Name:
Pre-image
A
3.
A
C
B
B
C
8.
F



D
C
C
E
 x
 y 


E
F
 x
 y 


Pre-image
B
C 



D
x  8 4 1 x 

y  7 10 3 y 
A
B
 x
 y 


x  8 4 1 x 

y  7 10 3 y 
A
6.
Image
x  8 4 1 x 

y  7 10 3 y 
A
5.
B
Translation



Translation
D E
x  4 6 9 7 5 x 

y  5 7 3 2 0  y 



Image
A

B
C
D
E
x
y 



Given each of the following pre-image triangles ABC and their image triangles ABC , fill in the
matrix equation that describes the translation that has occurred AND write a description (in words) of
this translation. You may wish to first graph the two triangles on your own graph paper.
1.
ABC has vertices A(6, –2), B(5, 4), C(1, 1); ABC has vertices A'(3, 2), B'(2, 8), C'(–2, 5).
Pre-image
A
x 

y 
B
Translation
Image
A
C
 x
 
 y 
 x
 
 y 
B
C



Description of the translation:
Matrices. 3.2
Rev S11
Mathematical Investigations II
Name:
2.
ABC has vertices A(–2, 7), B(1, 0), C(8, 4); ABC has vertices A'(–2, 5), B'(1, –2), C'(8, 2).
Pre-image
A
B
Translation
Image
A
C
x 

y 
 x
 
 y 
B
C
 x
 
 y 



Description of the translation:
3.
ABC has vertices A(3, 7), B(–8, 5), C(–2, –9); ABC has vertices A'(6, 5), B'(–5, 3),
C'(1, –11).
Pre-image
A
B
Translation
Image
A
C
x 

y 
 x
 
 y 
B
C
 x
 
 y 



Description of the translation:
4.
ABC has vertices A(–1, –1), B(–10, 6), C(2, –3); ABC has vertices A'(–10, 6), B'(–19, 13),
C'(–7, 4).
Pre-image
A
x 

y 
B
Translation
Image
A
C
 x
 
 y 
 x
 
 y 
B
C



Description of the translation:
Matrices. 3.3
Rev S11
Mathematical Investigations II
Name:
Given the pre-image polygons (the ones whose vertices are "non-primed" letters) graphed below and
their images (the ones whose vertices are "primed" letters), determine the translation matrix used for
each and write a matrix addition equation for each transformation.
5.
B'
B
A'
C'
A
C
6.
C'
D'
B'
C
D
B
E'
A'
E
A
Matrices. 3.4
Rev S11