Question

△abc is the image of △abc under a rotation about the origin, (0,0). 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 rules

Clockwise rotation:

• \(90^{\circ}\) clockwise: \((x,y)\to(y, -x)\)
• \(180^{\circ}\) clockwise: \((x,y)\to(-x,-y)\)
• \(270^{\circ}\) clockwise: \((x,y)\to(-y,x)\)

Counter - clockwise rotation:

• \(90^{\circ}\) counter - clockwise: \((x,y)\to(-y,x)\)
• \(180^{\circ}\) counter - clockwise: \((x,y)\to(-x,-y)\)
• \(270^{\circ}\) counter - clockwise: \((x,y)\to(y, -x)\)

Step2: Analyze the rotation

Let's assume a general point \(P(x,y)\) in \(\triangle ABC\) and its image \(P'(x',y')\) in \(\triangle A'B'C'\).
If we consider the rotation of a point from the blue triangle (\(\triangle ABC\)) to the red triangle (\(\triangle A'B'C'\)).
A \(270^{\circ}\) clockwise rotation is equivalent to a \(90^{\circ}\) counter - clockwise rotation.
For a \(270^{\circ}\) clockwise rotation: If we take a point \((x,y)\) in the original figure, after \(270^{\circ}\) clockwise rotation about the origin \((0,0)\), the new coordinates \((x',y')=(y, -x)\).
For a \(90^{\circ}\) counter - clockwise rotation: If we take a point \((x,y)\) in the original figure, after \(90^{\circ}\) counter - clockwise rotation about the origin \((0,0)\), the new coordinates \((x',y')=(y, -x)\)

Answer:

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