Question

△abc is the image of △abc under a rotation about point b.
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: Analyze rotation direction and angle

To determine the rotation angle, we observe the position of \( \triangle ABC \) and its image \( \triangle A'B'C' \) about point \( B \). A counterclockwise rotation of \( 90^\circ \) or a clockwise rotation of \( 270^\circ \) will map \( \triangle ABC \) to \( \triangle A'B'C' \). Let's verify:

• A \( 90^\circ \) counterclockwise rotation about \( B \): The vectors from \( B \) to \( A \) and \( B \) to \( C \) will rotate \( 90^\circ \) counterclockwise to match the vectors from \( B \) to \( A' \) and \( B \) to \( C' \).
• A \( 270^\circ \) clockwise rotation is equivalent to a \( 90^\circ \) counterclockwise rotation (since \( 360^\circ - 270^\circ = 90^\circ \) counterclockwise).

Step2: Evaluate each option

• Option A (\( 90^\circ \) clockwise): This would rotate the triangle in the opposite direction, not matching the image.
• Option B (\( 90^\circ \) counterclockwise): Matches the rotation as explained.
• Option C (\( 180^\circ \)): A \( 180^\circ \) rotation would flip the triangle, but the image does not match a \( 180^\circ \) rotation.
• Option D (\( 270^\circ \) clockwise): Equivalent to \( 90^\circ \) counterclockwise, so this also matches.
• Option E (\( 270^\circ \) counterclockwise): This is equivalent to \( 90^\circ \) clockwise, which does not match.

Answer:

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