Exploration Using GSP:

Coordinate Characterization of Linear Transformations


GSP problems involving equations for isometries. For each problem you should turn in a GSP
printout showing the result of your experiment, and a written explanation of what you did. (The
written explanation may be included as a GSP caption.)


PROBLEM 1

Using the GSP "Measure" menu and "Calculuate" submenu, verify the
equations for
(a) a translation,
(b) a rotation with center (0,0) and angle theta, and
(c) a reflection with mirror the line through (0,0) at angle theta with the x-axis.


Part A

We want to verify the following equation for a translation:

Translation by the vector (c,d):

x' = x + c
y' = y + d

Using GSP, Measure Coordinates, we measure the coordinates of point A=(x,y)=(-2.75, 1.79) and A"=(x',y')=(-1.35,0.42) along with the coordinates of the translation vector (c,d) = (1.40,-1.38). We obtain (c,d) by translating point O to the origin. In other words, by finding the change in x we find c, and by finding the change in y we find d. Next, we now know x and c, y and d, and can verify the given equation for a translation as shown below.


Part B

We want to verify the following equation:

Rotation with center (0,0) and angle theta:

x' = ax - by
y' = bx + ay

a = cos(theta), b = sin(theta)

 


Part C

We want to verify the following equation:

Reflection with mirror the line through (0,0) with angle theta:

x' = ax + by
y' = bx - ay

u = cos(theta), v = sin(theta)
a = u^2 - v^2, b = 2uv



PROBLEM 2

Using the same method, verify the equations for
(a) a rotation with arbitrary center and angle theta, and
(b) a reflection with arbitrary mirror.
(These equations are given in the book on pages 353-354. You can get them by combining the
equations of problem 1(b)(c) with translations.)


PART A

We want to verify the equation of the Coordinate Form of Rotation about Center(h,k):

(x' - h) = (x - h) cos (theta) - (y - k) sin (theta)

(y' - k) = (x - h) sin (theta) + (y - k) cos (theta)

where theta = measure of angle of rotation

 


PART B

We want to verify the equation for Coordinate Form of Reflection in a Line

x' = (b^2 - a^2) x - 2aby + 2ac

y' = -2abx - (b^2 - a^2) y + 2bc


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