Name __________________
Worksheet 10.1
Transformations I
Draw the triangle with vertices A  (1, 1), B  (1, 3), and C  (5, 3) . For this entire worksheet, this
will be the triangle that you will transform.
1. Rotation about the Origin
(a) Rotate ABC 90 counter-clockwise about
the origin and draw this new triangle. In other
words, draw the image G90 (ABC)  A'B'C'
(b) If you move clockwise around A'B'C' , are
the vertices still in the same order?
(c) Is the image A'B'C' similar to the original
ABC ? Is it congruent?
(d) Find the ratio
area A 'B'C '
area ABC
(e) Perform the same 90 counter-clockwise
1 
0 
rotation on the vectors ˆi    and ˆj   
0 
1 
(f) Write down a 2  2 matrix such that the first column is the image of î and the second column
is the image of ˆj
(g) Convert the points A  (1, 1) , B  (1, 3) , and C  (5, 3) into vectors that originate at the origin
(i.e. make each one a column vector)
(h) Use the matrix you got in part (f) to transform each of these vectors.
(i) Interpret your result.
2. Reflection through the Origin
(a) Draw RO (ABC)  A'B'C' and
RO (iˆ) and RO ( ˆj )
(b) If you move clockwise around A'B'C' , are
the vertices still in the same order?
(c) Is the image A'B'C' similar to the original
ABC ? Is it congruent?
(d) Find the ratio
area A 'B'C '
area ABC
(e) Find the 2  2 transformation matrix for
reflection through the origin.
(f) Find the coordinates of A', B', and C ' using matrix multiplication
3. Reflection across the Line y = x
(a) Draw RL (ABC)  A'B'C' and
RL (iˆ) and RL ( ˆj ) where L is the line y = x
(b) If you move clockwise around A'B'C' , are
the vertices still in the same order?
(c) Is the image A'B'C' similar to the original
ABC ? Is it congruent?
(d) Find the ratio
area A 'B'C '
area ABC
(e) Find the 2  2 transformation matrix for
reflection across the line y = x.
(f) Find the coordinates of A', B', and C ' using matrix multiplication.
4. Reflection over x-axis
(a) Draw Rxaxis (ABC)  A'B'C' and
Rxaxis (iˆ) and Rxaxis ( ˆj )
(b) If you move clockwise around A'B'C' , are
the vertices still in the same order?
(c) Is the image A'B'C' similar to the original
ABC ? Is it congruent?
(d) Find the ratio
area A 'B'C '
area ABC
(e) Find the 2  2 transformation matrix for
reflection over the x-axis.
(f) Find the coordinates of A', B', and C ' using matrix multiplication
5. Dilation by a Factor of 2
(a) Draw D2 (ABC)  A'B'C' and
D2 (iˆ) and D2 ( ˆj ) This is a dilation about the
origin.
(b) If you move clockwise around A'B'C' , are
the vertices still in the same order?
(c) Is the image A'B'C' similar to the original
ABC ? Is it congruent?
(d) Find the ratio
area A 'B'C '
area ABC
(e) Find the 2  2 transformation matrix for
dilation by a factor of 2 about the origin.
(f) Find the coordinates of A', B', and C ' using matrix multiplication.
6. Importance of the determinant.
(a) Find the determinant of the transformation matrix (the 2  2 matrix) from part 2
(Reflection through origin)
(b) Find the determinant of the transformation matrix from part 3
(Reflection across a line)
(c) Find the determinant of the transformation matrix from part 4
(Also a reflection across a line)
(d) Find the determinant of the transformation matrix from part 5
(Dilation)
(e) Relate the determinant’s sign and magnitude to its effect on the transformed triangle
A'B'C' . How is it related to the area ratio? How about the relative orientation?
(f) What is area of A'B'C' if its transformation matrix has a determinant of 9?
7. Using Transformation Matrices
Now perform the following two transformations
on triangle ABC. First reflect it across the line
y = x and then reflect it across the y axis.
(a) What are the coordinates of the resulting
triangle?
(b) What single transformation accomplishes the
same thing?
(c) Find the transformation matrix for reflection across the line y = x.
(d) Find the transformation matrix for reflection across the y-axis.
(e) Multiply these two matrices and interpret the result. Does the order in which you place the
two matrices matter? Show your work.
8. A Few other Transformations
(a) Explain why it is impossible to represent translation of a point using our current method of
matrix multiplication? (Hint: think about what happens to the origin)
(b) Find a single matrix that dilates by a factor of 3, but only in the x-direction (see section 9.2
and theorem 9-6 in the book for a description of such a dilation).
(c) Find a single matrix that performs the transformation G45
1 0 
(d) Describe the effect of the matrix 
 (See page 266 in book for a hint)
4 1
(e) In terms of  , give the 2  2 matrix for the general rotation G (You will need to use trig
functions)