Multiple Choice

1. A point P is located at (4, -1). This point undergoes a counterclockwise rotation of 90 degrees about the origin, followed by a translation defined by the vector $\langle -5, 3 \rangle$. What are the final coordinates of the image P'?

   (-4, 7)

   (-1, 2)

   (1, 7)

   (6, 7)

2. Which 2x2 matrix, when multiplied by a coordinate vector $\begin{pmatrix} x \\ y \end{pmatrix}$, represents a counterclockwise rotation of 270 degrees about the origin?

   $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

   $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

   $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

   $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

3. A polygon with an area of 12 square units is transformed by the dilation matrix $M = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$. What is the area of the transformed polygon?

   300 square units

   60 square units

   120 square units

   30 square units

4. The matrix $R = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ is applied to all vertices of a figure. Which geometric transformation does this matrix represent?

   Reflection across the x-axis

   Reflection across the line $y = x$

   Rotation of 180 degrees

   Reflection across the y-axis

5. A figure is subjected to two sequential transformations: (1) Rotation of $90^\circ$ counterclockwise ($R_{90}$) and (2) Translation by $T = \langle 1, 0 \rangle$. If the order is reversed (T followed by $R_{90}$), how does the final image differ from the original final image?

   The final image would have the same orientation but would be shifted vertically by one unit.

   The final image would be rotated by $90^\circ$ but translated back one unit in the x-direction.

   The final image would be exactly the same, as rotations and translations commute.

   The final image would have the same orientation but would be shifted right by one unit and down by one unit.

6. A single transformation matrix $M$ is applied to a figure, resulting in coordinates $(x', y')$ where $x' = -3x$ and $y' = -3y$. This matrix $M$ represents a composite transformation of:

   Rotation of 90 degrees followed by Dilation by scale factor 3

   Reflection across the y-axis followed by Dilation by scale factor 3

   Dilation by scale factor 3 followed by Rotation of 180 degrees

   Reflection across the x-axis followed by Rotation of 180 degrees

7. A triangle undergoes a rotation of 180 degrees about the origin. What single rotation must be applied to the image to return it exactly to the original position?

   A rotation of 90 degrees clockwise

   A rotation of 360 degrees clockwise

   A rotation of 90 degrees counterclockwise

   A rotation of 180 degrees

8. A figure is transformed by the matrix $M = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$. This matrix represents a reflection across which line?

   The y-axis

   The x-axis

   The line $y = -x$

   The line $y = x$

9. A coordinate pair $(x, y)$ is multiplied by the matrix $M = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$. What is the geometric effect of this transformation?

   The figure is translated to the origin.

   The figure is rotated 180 degrees.

   Every point maps to the origin (0, 0).

   The figure collapses onto the x-axis.

10. A point P is rotated $90^\circ$ counterclockwise to obtain the image P' at (-7, 2). What were the original coordinates of point P?

    (2, 7)

    (7, -2)

    (-2, -7)

    (2, -7)

11. Which of the following 2x2 matrices represents a transformation that is NOT an isometry (a rigid motion)?

    $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ (Reflection)

    $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ (Rotation)

    $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ (Reflection)

    $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ (Stretch)

12. The transformation rule defined by $(x, y) \rightarrow (4x + 1, 4y - 3)$ combines which operations?

    Reflection and Dilation

    Dilation and Translation

    Rotation and Translation

    Rotation and Dilation

13. What single rotation is equivalent to performing a rotation of $180^\circ$ clockwise followed immediately by a rotation of $90^\circ$ counterclockwise?

    $270^\circ$ clockwise rotation

    $45^\circ$ counterclockwise rotation

    $90^\circ$ counterclockwise rotation

    $90^\circ$ clockwise rotation

14. If the determinant of a 2x2 transformation matrix is 4, what does this value indicate about the resulting image compared to the original figure?

    The perimeter of the figure has been scaled by a factor of 4.

    The figure's orientation has been reversed (a reflection occurred).

    The transformation must include a 180-degree rotation.

    The area of the figure has been scaled by a factor of 4.

15. Point A (3, 5) undergoes a reflection across the line $y=x$, and then is translated by $T = \langle -1, -1 \rangle$. What are the final coordinates of A''?

    (2, 4)

    (4, 2)

    (4, 6)

    (6, 4)

16. A transformation involves a reflection across the x-axis, followed by a dilation with a scale factor of 2. Which single 2x2 matrix represents this composite transformation?

    $\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$

    $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} -2 & 0 \\ 0 & 2 \end{pmatrix}$

    $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$

17. A figure is transformed by the matrix $M = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$. This transformation is equivalent to which single rotation?

    Reflection across the x-axis

    Rotation of $90^\circ$ clockwise

    Rotation of $270^\circ$ counterclockwise

    Rotation of $180^\circ$

18. A rectangle centered at (1, 2) is transformed first by rotation $R_{180}$, then by a dilation factor of 2, and finally translated by $\langle 0, -5 \rangle$. Where is the new center of the rectangle?

    (-2, -9)

    (-2, -5)

    (2, 9)

    (-2, -1)

19. Which composition of geometric transformations, T (Translation) and $R_{90}$ (Rotation of $90^\circ$), will always produce the same result regardless of the order they are applied (i.e., they are commutative)?

    None of the above; matrix transformations involving rotation and translation are generally non-commutative.

    Rotation of $90^\circ$ followed by Translation by $\langle 0, 0 \rangle$.

    Translation by $\langle 1, 1 \rangle$ followed by Translation by $\langle 2, 2 \rangle$.

    Reflection across the x-axis followed by reflection across the y-axis.

20. Consider the general rotation matrix $R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. If a rotation matrix is $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$, what is the smallest positive angle $\theta$ of rotation?

    $360^\circ$

    $90^\circ$

    $180^\circ$

    $270^\circ$