Lab 3

The goal of this lab is to

• Find matrix representations for the symmetries of a square
• Find matric representations for more general line reflections and rotations.
• Find a general area formula related to matrices

You will work with one other person, who will be your partner on Friday, also. You will write and turn in a short report on your results today, as explained below.

Part 1 (GSP technique on plotting)

In a new sketch, choose Graph>Define Coordinate System. You will see a coordinate system and a draggable origin and unit point on the x-axis.

Practice plotting with parameters

• Use Graph>New Parameter to create two parameters with values 2 and 3.
• Then select the two parameters and choose Graph>Plot as (x,y).
• Next, select the two parameters in the opposite order and Plot as (x,y) again. If you chose the different order, you should have two points.
• Change one or both of the parameters and see how the points change. You can animate a parameter also. How are the points related?

Practice plotting with measurements and calculations

• In the same sketch, draw a point A. Then measure the x and y coordinates by Measure>Abscisse and Measure>Ordinate.
• Now choose the measurements with y(A) first and then x(A). Plot as (x,y). How are the points related.
• Next, use the calculator to compute -x(A) and -y(A). Plot both (-x, -y) and (-y,-x). How are these points related to A?

Part 2 (Symmetries of a Square)

• In a new sketch, define a coordinate system and then construct by plotting or by usual construction methods a square with sides parallel to the axes and one vertex at (1,1).
• Construct the lines of symmetry of this square.
• Draw a free point A and measure x = x(A) and y = y(A). Also calculate -x and -y.

The goal of this section is plot various combinations of x, y, -x, -y to get the image of A by all 8 symmetries of the square (A itself is one of these, so you only have 7 to go). IMPORTANT. These points should be plotted not constructed using the Transform menu!

Then for each of the 8 symmetries, write for turning in a short report with this:

• tell what the symmetry is
• record the formula for each and also the 2x2 matrix.
• If e1 = (1,0) and e2 = (0,1), tell what is the image of e1 and e2 (write as column vectors)
• Put the two column vectors together to get the matrix of the transformation.

Part 3 (Rotation and Reflection Matrix in general)

• In a new sketch, define a coordinate system and construct the unit circle.
• Also construct a point M on the unit circle.Measure the x and y coordinates of M and name them m and n, so M = (m,n).
• Let E = the unit e1 = (1,0) and O = (0,0). Then the rotation with center O and angle EOM maps E to M. What are the coordinates of the image of e2?
• Put this together to write down the matrix of this rotation.
• If the angle EOM = t, what are m and n in terms of trigonometric functions of t?
• Rewrite the matrix with trig functions.

The same reasoning works for reflections.

• In the same sketch, M is the image of the reflection of e1 in the angle bisector of EOM. What is the image of e2 by this reflection? What is the matrix of this reflection.

(Harder: What is the matrix of the reflection in the line OM?)

Part 4 (Area of a triangle)

• In this new sketch, we will define a coordinate system with units = cm. Start with a new parameter of 1 cm. Then select the parameter and choose Graph>Define Unit Distance.
• Draw points A = (a,b) and C = (c,d). Then construct a point D so that OADC is a parallelogram.
• Next, construct the point C' of intersection of line DC and the y-axis. Construct a point D so that OAD'C' is a parallelogram.
• Explain why this new parallelogram has the same area as OADC.
• Next, construct the point A' of intersection of line D'A and the x-axis. Construct a point D'' so that OA'D'C' is a parallelogram. Explain why this new parallelogram has the same area as OADC.
• In terms of a, b, c, d, what are the coordinates of C' and A'? What is the area of OA'D'C' in terms of a, b, c, d?

Now explain what happens when A and C are moved so that they are approximately reversed (something about the sign of your formula).

Extra: Jigsaw puzzle. If A and C are in the first quadrant, draw a rectangle with two sides on the axes and the other sides as small as possible so that the parallelogram OADC is inside the rectangle. Break up the area between the parallelogram and the rectangle into pieces such as triangles and trapezoid whose areas you can compute. Use this to compute the area of the parallelogram.