Question

a rectangle is transformed according to the rule ( r_{0,90^{circ}} ). the image of the rectangle has vertices located at ( r(-4,4) ), ( s(-4,1) ), ( p(-3,1) ), and ( q(-3,4) ). what is the location of ( q )?
( (-4,-3) )
( (-3,-4) )
( (3,4) )
( (4,3) )

Explanation:

Step1: Recall the rotation rule

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin \(R_{0,90^{\circ}}\) is \((x,y)\to(-y,x)\). To find the pre - image, we use the inverse rule \((x,y)\to(y, - x)\) for a \(90^{\circ}\) clockwise rotation (since rotation \(R_{0,90^{\circ}}\) and its inverse are related).

Step2: Apply the inverse rotation rule to \(Q'(-3,4)\)

Let \((x',y')=(-3,4)\). Using the formula \((x,y)=(y',-x')\).
Substitute \(x'=-3\) and \(y' = 4\) into the formula:
\(x = 4\) and \(y=-(-3)=3\)

Answer:

\((4,3)\)