Geometric rotations are transformations that turn a figure around a fixed point, known as the center of rotation. In the coordinate plane, the origin (0, 0) is often used as this center. Common rotations include 90, 180, and 270 degrees. Each rotation follows a specific algebraic rule: for a 90-degree clockwise rotation, (x, y) becomes (y, -x); for a 180-degree rotation, (x, y) becomes (-x, -y); and for a 270-degree clockwise rotation (or 90-degree counterclockwise), (x, y) becomes (-y, x). These transformations preserve the size and shape of the figure while changing its orientation and position within the quadrants.
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A 180-degree rotation (clockwise or counterclockwise) about the origin changes the signs of both the x and y coordinates, following the rule \( (x, y) \rightarrow (-x, -y) \).
Triangle A is rotated 90° counterclockwise with the origin as the center of rotation to create a new figure. Which triangle shows the new location?
Explanation
A 90-degree counterclockwise rotation moves a figure from one quadrant to the next in a counterclockwise direction. Triangle A is in Quadrant II; rotating it 90 degrees counterclockwise places it in Quadrant III, where Triangle D is located.
What is the image of S (2, -5) after a rotation of 90° clockwise about the origin?
Explanation
The rule for a 90-degree clockwise rotation about the origin is \( (x, y) \rightarrow (y, -x) \). Applying this to S(2, -5), the image is S'(-5, -(2)), which is S'(-5, -2).
If you were to rotate ABCD 180° about the origin, what would the coordinate of A' be?
Explanation
The coordinate of vertex A is (3, 5). A 180-degree rotation follows the rule \( (x, y) \rightarrow (-x, -y) \). Therefore, the coordinate of A' is (-3, -5).
What is the image of W (-5, -3) after a rotation 270° counterclockwise?
Explanation
A 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. The rule is \( (x, y) \rightarrow (y, -x) \). Applying this to W(-5, -3), we get W'(-3, -(-5)), which is W'(-3, 5).
By checking the coordinates of vertex E (-2, 1) and its image E' (1, 2), we see the transformation follows the rule \( (x, y) \rightarrow (y, -x) \), which corresponds to a 90-degree clockwise rotation.
What transformation is happening? \( (x, y) \rightarrow (x + 2, y - 3) \)
Explanation
In coordinate transformations, adding to the x-coordinate (x + 2) moves the figure right, and subtracting from the y-coordinate (y - 3) moves it down. This is called a translation or 'slide'.
Given the parallelogram PQRS with points P(-2, 4), Q(0, 3), R(-2, 2), and S(-4, 3), which graph correctly shows parallelogram P'Q'R'S' after a 180° clockwise rotation?
Explanation
A 180-degree rotation maps (x, y) to (-x, -y). Thus, P(-2, 4) maps to P'(2, -4) and Q(0, 3) maps to Q'(0, -3). Graph (a) is the only one where the image vertices are correctly positioned in the fourth quadrant.
Under a rotation about the origin, the point (-3, 4) is mapped to the point (3, -4). What is the measure of the angle of rotation?
Explanation
The transformation from (-3, 4) to (3, -4) changes the signs of both coordinates, which follows the rule \( (x, y) \rightarrow (-x, -y) \). This rule defines a 180-degree rotation.
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