Download 6.4 Dilations

Mathematical Investigations II
Name:
Mathematical Investigations II
MATRICES & GEOMETRIC TRANSFORMATIONS
Transformations 3: Size Transformations and Scalar Multiplication
1.
Consider ABC and A'B'C'
graphed at the right:
a.
y
Write the vertex matrix for
ABC and the vertex matrix
for A'B'C'
C'
C
 _____ _____ _____ 
 _____ _____ _____ 


vertex matrix ABC
B'
B
x
A
 _____ _____ _____ 
 _____ _____ _____ 


vertex matrix ABC 
A'
b.
Describe the similarities and differences between the numbers in the vertex matrix for ABC
and the vertex matrix for A'B'C'.
c.
Describe the geometric transformation that “maps” ABC onto A'B'C'.
d.
Is there a translation matrix that “maps” ABC onto A'B'C'? If so, determine the translation
matrix and write a matrix addition equation for this transformation. If not, explain why?
Matrices 6.1
Rev S11
Mathematical Investigations II
Name:
2.
Consider ABC and A'B'C'
graphed at the right:
a.
y
Write the vertex matrix for
ABC and the vertex matrix
for A'B'C'
B
B'
 _____ _____ _____ 
 _____ _____ _____ 


vertex matrix ABC
x
C'
 _____ _____ _____ 
 _____ _____ _____ 


vertex matrix ABC 
C
A'
A
b.
Describe the similarities and differences between the numbers in the vertex matrix for ABC
and the vertex matrix for A'B'C'.
c.
Describe the geometric transformation that “maps” ABC onto A'B'C'.
d.
Is there a translation matrix that “maps” ABC onto A'B'C'? If so, determine the translation
matrix and write a matrix addition equation for this transformation. If not, explain why?
Matrices 6.2
Rev S11
Mathematical Investigations II
Name:
You should have noticed that in mapping ABC onto A'B'C', all of coordinates were doubled in the first
problem and all of the coordinates were halved in the second problem. In addition, it should have been
clear that there is no translation matrix that maps ABC onto A'B'C' because the two triangles are of
different sizes (though they do have the same shape; such triangles are called similar triangles). This
type of transformation is called a dilation. The dilation in problem #1 is sometimes referred to as D2
since the coordinates of the image triangle, A'B'C', are all double those of the original triangle, ABC.
Similarly, the dilation in problem #2 is sometimes referred to as D1 .
2
So now the question is, what can we do to the vertex matrix for ABC to change it into the vertex matrix
for A'B'C'? The answer involves multiplying a matrix by a number (or scalar), which is called scalar
matrix multiplication (or simply scalar multiplication).
Scalar matrix multiplication works a little like the distributive property. When you multiply a matrix by
a scalar, you simply multiply each element of the matrix by that scalar. For example,
9 6
 1 3 2   3
3 



5 4 8 15 12 24
3.
Complete the following for ABC and its image A'B'C' from problem #1:



scalar
4.



 




vertex matrix for ABC
vertex matrix for ABC 
Complete the following for ABC and its image A'B'C' from problem #2:



scalar



 




vertex matrix for ABC
vertex matrix for ABC 
Matrices 6.3
Rev S11