Question

a triangle is rotated 90° about the origin. which rule describes the transformation? (x,y)→(−x,−y) (x,y)→(−y,x) (x,y)→(−y,−x) (x,y)→(y,−x)

Explanation:

Step1: Recall rotation rules

When a point \((x,y)\) is rotated \(90^{\circ}\) counter - clockwise about the origin, the new coordinates \((x',y')\) can be found using the rotation matrix \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

\).

Multiplying the matrices: \(x' = 0\times x+(- 1)\times y=-y\) and \(y'=1\times x + 0\times y=x\). So the rule is \((x,y)\to(-y,x)\).

Step2: Check other options

• For \((x,y)\to(-x,-y)\), this is a \(180^{\circ}\) rotation about the origin.
• For \((x,y)\to(-y,-x)\), this is a \(270^{\circ}\) counter - clockwise (or \(90^{\circ}\) clockwise) rotation about the origin.
• For \((x,y)\to(y,-x)\), this is a \(270^{\circ}\) counter - clockwise (or \(90^{\circ}\) clockwise) rotation about the origin if we consider the standard rotation rules.

Answer:

\((x,y)\to(-y,x)\) (the second option)