Which transformation would change the coordinates of vertex C to (2, −3)?
A.
a translation 4 units down and 3 units to the left
B.
a reflection across the x-axis
C.
a 90° counterclockwise rotation about the point (3, 2)
D.
a 90° clockwise rotation about the origin
The correct answer is:
B. a reflection across the x-axis
A reflection across the x-axis will change the y-coordinate of vertex C from positive to negative, which will result in the y-coordinate being -3. The x-coordinate will remain the same.
To change the coordinates of vertex C to (2, -3), you would need to perform a transformation.
Let's analyze each transformation option:
A. Translation: This option moves the vertex downward and leftward by specific amounts. However, it does not guarantee that the new coordinates will be (2, -3). Therefore, option A is not the correct transformation.
B. Reflection across the x-axis: This option reflects the vertex across the x-axis, which changes the sign of the y-coordinate. However, it does not change the x-coordinate and cannot result in (2, -3). Therefore, option B is not the correct transformation.
C. 90° counterclockwise rotation about the point (3, 2): This option rotates the vertex 90° counterclockwise around the given point. While rotating counterclockwise would change the coordinates, it does not necessarily result in (2, -3). Therefore, option C is not the correct transformation.
D. 90° clockwise rotation about the origin: This option rotates the vertex 90° clockwise around the origin. Since the question does not provide the original coordinates of vertex C, it is uncertain whether this option will produce the desired result. Therefore, option D cannot be confirmed as the correct transformation.
Based on the given options, there is no transformation that can be determined as a definitive answer to change the coordinates of vertex C to (2, -3).
To determine which transformation would change the coordinates of vertex C to (2, -3), we need to analyze the given options and understand how each transformation affects the coordinate points.
A. Translation: This transformation shifts the figure horizontally and vertically without changing its orientation. A translation 4 units down and 3 units to the left would not change the x-coordinate of vertex C to 2.
B. Reflection across the x-axis: This transformation reflects the figure across the x-axis, changing the sign of the y-coordinate. However, reflecting vertex C across the x-axis would yield the coordinates (−1, 3), not (2, −3).
C. 90° counterclockwise rotation about the point (3, 2): This transformation rotates the figure 90° counterclockwise around the given point. To perform this rotation, we can find the vector connecting the point of rotation (3, 2) to vertex C, and then rotate this vector 90° counterclockwise.
The vector from (3, 2) to C is:
(x2 - x1, y2 - y1) = (3 - (-1), 5 - 2) = (4, 3)
Rotating this vector 90° counterclockwise gives us the new vector:
(-y, x) = (-3, 4)
Adding this vector to the point of rotation (3, 2), we get the coordinates of the new point:
(3 - 3, 2 + 4) = (0, 6)
Thus, the 90° counterclockwise rotation about the point (3, 2) would not change the coordinates of vertex C to (2, −3).
D. 90° clockwise rotation about the origin: This transformation rotates the figure 90° clockwise around the origin (0, 0). To perform this rotation, we can find the vector from the origin to vertex C, and then rotate this vector 90° clockwise.
The vector from the origin to C is:
(x2 - x1, y2 - y1) = (-1 - 0, 5 - 0) = (-1, 5)
Rotating this vector 90° clockwise gives us the new vector:
(y, -x) = (5, 1)
Adding this vector to the origin (0, 0), we get the coordinates of the new point:
(0 + 5, 0 - 1) = (5, -1)
Thus, the 90° clockwise rotation about the origin would not change the coordinates of vertex C to (2, −3).
Therefore, none of the given transformations would change the coordinates of vertex C to (2, −3).