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# Matrices and Transforming Geometric Shapes (Grades 11-12)

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## Matrices and Transforming Geometric Shapes

1.
Given the triangle with vertices $(1,1), (2,3),$ and $(5,1)$, which of the following matrix expressions would represent the reflection of this triangle over the y-axis?
1. $[[1,0],[0,-1]] \ [[1,1],[2,3],[5,1]]$
2. $[[-1,0],[0,1]] \ [[1,2,5],[1,3,1]]$
3. $[[-1,0],[0,1]] \ [[1,1],[2,3],[5,1]]$
4. $[[1,0],[0,-1]] \ [[1,2,5],[1,3,1]]$
2.
The vertex matrix for rectangle ABCD is $V = [[1,1,4,4],[1,3,3,1]]$. Which of the following is the correct transformed vertex matrix, if the transformation matrix $A = [[0,1],[-1,0]]$ is applied?
1. $[[1,-1],[3,-1],[3,-4],[1,-4]]$
2. $[[-1,-3,-3,-1],[1,1,4,4]]$
3. $[[1,3,3,1],[-1,-1,-4,-4]]$
4. $[[-1,-3,-3,-1],[-1,-1,-4,-4]]$
3.
For the vertex matrix $V = [[1,1,3,3],[-1,2,2,-1]]$ and transformation matrix $M = [[1,2],[0,1]]$, what is the resulting shape if M is applied to V? Choose the most specific classification possible.
1. Square
2. Parallelogram
3. Rectangle
4.
Put the vertices of the shape in the graph below into a vertex matrix, and then apply the rotation matrix $A = [[sqrt(2)/2, -sqrt(2)/2],[sqrt(2)/2, sqrt(2)/2]]$. By what angle, measured in degrees, has the shape been rotated (given that the angle of rotation is counterclockwise)?
1. 30°
2. 45°
3. 60°
4. 105°
5.
Which of the following equations correctly state that, for a figure in the cartesian plane, a reflection over the x-axis, followed by a reflection over the y-axis, is identical to a 180° rotation (either clockwise or counterclockwise)?
1. $[[1,0],[0,1]] [[-1,0],[0,-1]] = [[-1,0],[0,-1]]$
2. $[[1,0],[0,-1]] [[-1,0],[0,1]] = [[0,1],[1,0]]$
3. $[[-1,0],[0,1]] [[1,0],[0,-1]] = [[-1,0],[0,-1]]$
4. $[[1,0],[0,-1]] [[-1,0],[0,1]] = [[-1,0],[0,-1]]$
6.
Which one of the following matrix equations correctly shows the transformation of the first rectangle, whose vertex matrix is $V_1 = [[1,1,4,4],[1,2,2,1]]$, to the second rectangle, whose vertex matrix is $V_2 = [[-4,-4,2,2],[1,3,3,1]] ?$

1. $V_2 = [[2,0],[0,2]] (V_1 - [[1,1,1,1],[1,1,1,1]] ) - [[4,4,4,4],[-1,-1,-1,-1]]$
2. $V_2 = [[2,0],[0,2]] V_1 - [[5,5,5,5],[0,0,0,0]]$
3. $V_2 = [[-4,1/2],[3/2, 1]] V_1$
4. $V_2 = [[2,2],[2,2]] ( V_1 - [[1,1,1,1],[1,1,1,1]])$
7.
For the following questions, let $T = [[2,-2],[0,3]]$ be a transformation matrix. Also, let $V_1 = [[2,5,5,2],[2,2,-1,-1]]$ be the vertex matrix of a square whose area is 9 units squared.
A.
What is the resulting matrix if $V_1$ is transformed by $T$? Let this matrix be $V_2$.
1. $[[0,6,12,6],[6,6,-3,-3]]$
2. $[[0,6,5,2],[6,6,-1,-1]]$
3. $[[2,6,8,2],[0,6,3,-1]]$
4. $[[4,10,10,4],[6,6,-3,-3]]$
B.
What is the area of the resulting shape, defined by the vertex matrix $V_2 ?$
1. 108 units squared
2. 72 units squared
3. 54 units squared
4. 36 units squared
C.
What is the absolute value of the determinant of the transformation matrix, $T ?$
1. 6
2. 7
3. 12
4. 3
D.
How does the absolute value of the determinant of the transformation matrix relate to the area of the two quadrilaterals (the square, $S_1$ and the transformed shape, $S_2$)? Choose the equation which correctly describes this relationship.
1. $|det(T)| = "Area"(S_1) \ "Area"(S_2)$
2. $"Area"(S_2) = |det(T)| \ "Area"(S_1)$
3. $|det(T)| = |"Area"(S_2) - "Area"(S_1)|$
4. There is no relationship between the absolute value of the determinant of T and the area of the shapes.
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