Question

△abc is the image of △abc under a rotation about point p.
determine the angles of rotation.
choose all answers that apply:
a 90° clockwise
b 90° counterclockwise
c 180°
d 270° clockwise
e 270° counterclockwise

Explanation:

Step1: Recall Rotation Properties

To determine the rotation angle, we analyze the position of \( \triangle ABC \) and its image \( \triangle A'B'C' \) about point \( P \). A \( 90^\circ \) counterclockwise rotation or a \( 270^\circ \) clockwise rotation will map \( \triangle ABC \) to \( \triangle A'B'C' \) (and vice - versa for the other pair). Let's consider the direction of rotation.

• A counterclockwise rotation of \( 90^\circ \) (or a clockwise rotation of \( 270^\circ \)) will align the pre - image and the image.
• A clockwise rotation of \( 90^\circ \) is equivalent to a counterclockwise rotation of \( 270^\circ \), and a \( 180^\circ \) rotation would result in a different orientation (the image would be opposite in both x and y directions relative to the pre - image, which is not the case here).

Step2: Analyze Each Option

• Option A (\( 90^\circ \) clockwise): If we rotate \( \triangle ABC \) \( 90^\circ \) clockwise about \( P \), the resulting triangle will not match \( \triangle A'B'C' \).
• Option B (\( 90^\circ \) counterclockwise): Rotating \( \triangle ABC \) \( 90^\circ \) counterclockwise about \( P \) will map it to \( \triangle A'B'C' \).
• Option C (\( 180^\circ \)): A \( 180^\circ \) rotation would flip the triangle in both horizontal and vertical directions, and the resulting triangle will not match \( \triangle A'B'C' \).
• Option D (\( 270^\circ \) clockwise): A \( 270^\circ \) clockwise rotation is equivalent to a \( 90^\circ \) counterclockwise rotation. So rotating \( \triangle ABC \) \( 270^\circ \) clockwise about \( P \) will also map it to \( \triangle A'B'C' \).
• Option E (\( 270^\circ \) counterclockwise): A \( 270^\circ \) counterclockwise rotation is equivalent to a \( 90^\circ \) clockwise rotation, which will not map \( \triangle ABC \) to \( \triangle A'B'C' \).

Answer:

B. \( 90^\circ \) counterclockwise, D. \( 270^\circ \) clockwise