Question

describe the rotation as a (a) counterclockwise rotation and (b) clockwise rotation.
a. 180 degrees counterclockwise rotation about the origin
b. 90 degrees clockwise rotation about the origin
write a rule for the rotation.
$(x,y)\to(\square,\square)$

Explanation:

Step1: Recall rotation rules

For a \(90^{\circ}\) clockwise rotation about the origin, the rule is \((x,y)\to(y, -x)\). For a \(180^{\circ}\) counter - clockwise rotation about the origin, the rule is \((x,y)\to(-x,-y)\). But since we can also describe the rotation as a \(90^{\circ}\) clockwise rotation (which gives the same result as a \(270^{\circ}\) counter - clockwise rotation), and we want a single rule.
Let's use the general rotation formula. If we consider the rotation from the blue triangle to the red triangle.
Take a point, say \(A(2,0)\). After rotation, \(A'\) is \((0,2)\) (if we consider the transformation as a \(90^{\circ}\) clockwise rotation).
The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y, -x)\).

Answer:

\((x,y)\to(y, -x)\)