Question

quadrilateral abcd is transformed according to the rule $(x,y)\to(y, -x)$. which is another way to state the transformation? $r_{0,90^{circ}}$ $r_{0,180^{circ}}$ $r_{0,270^{circ}}$ $r_{0,360^{circ}}$

Explanation:

Step1: Recall rotation rules

• For a rotation of \( 90^{\circ} \) counter - clockwise about the origin \( R_{0,90^{\circ}}:(x,y)\to(-y,x) \).
• For a rotation of \( 180^{\circ} \) about the origin \( R_{0,180^{\circ}}:(x,y)\to(-x,-y) \).
• For a rotation of \( 270^{\circ} \) counter - clockwise about the origin (or \( 90^{\circ} \) clockwise), using the rotation matrix \(

$$\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$

\) with \( \theta = 270^{\circ}\), \(\cos270^{\circ}=0\), \(\sin270^{\circ}=- 1\). If we have a point \((x,y)\), the new coordinates \((x',y')\) are given by \(x'=x\cos270^{\circ}-y\sin270^{\circ}=y\) and \(y'=x\sin270^{\circ}+y\cos270^{\circ}=-x\), so \(R_{0,270^{\circ}}:(x,y)\to(y, - x)\).

• For a rotation of \( 360^{\circ} \) about the origin \( R_{0,360^{\circ}}:(x,y)\to(x,y) \).

Since the transformation rule is \((x,y)\to(y,-x)\), it matches the rule for a \( 270^{\circ} \) counter - clockwise rotation about the origin.

Answer:

\( R_{0,270^{\circ}} \)