Question

figure pqrs is rotated around the origin to form figure pqrs. which rotation is shown in the coordinate plane? 90° counterclockwise 180° clockwise 90° clockwise

Explanation:

Step1: Recall rotation rules

• For a \(90^{\circ}\) counter - clockwise rotation \((x,y)\to(-y,x)\)
• For a \(180^{\circ}\) rotation \((x,y)\to(-x,-y)\)
• For a \(90^{\circ}\) clockwise rotation \((x,y)\to(y, - x)\)

Step2: Assume a point

Let's assume a point \(P\) (say \(P(-3,1)\)). After rotation, if it's \(90^{\circ}\) clockwise, the new coordinates \((x',y')=(1,3)\) (not matching). If it's \(180^{\circ}\) rotation \((x',y')=(3, - 1)\) (not matching). If it's \(90^{\circ}\) clockwise, for a general point \((x,y)\) of the figure \(PQRS\), using the rule \((x,y)\to(y,-x)\).

Let's take another approach: visually, the direction of rotation. A \(90^{\circ}\) clockwise rotation turns the figure in the direction of a clock's hand movement (to the right - down). A \(90^{\circ}\) counter - clockwise is to the left - up. A \(180^{\circ}\) rotation flips the figure halfway around.

By observing the orientation of \(PQRS\) and \(P'Q'R'S'\), the rotation is \(90^{\circ}\) clockwise.

Answer:

C. \(90^{\circ}\) clockwise