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 )?
options:
( circ (-4, -3) )
( circ (-3, -4) )
( circ (3, 4) )
( circ (4, 3) )

Explanation:

Step1: Recall rotation rule for 90° origin

90° counterclockwise rotation: $(x,y) \to (-y,x)$
90° clockwise rotation: $(x,y) \to (y,-x)$

Step2: Find rotation using P and P'

Given $P'(x',y') = (-3,1)$ (image of P). Test 90° clockwise:
If $P'(y,-x) = (-3,1)$, then $y=-3$, $-x=1 \Rightarrow x=-1$. (Not needed for Q)

Step3: Apply inverse rotation to Q'

Q' = (-3,4). Inverse of 90° clockwise is 90° counterclockwise:
Inverse rule: $(x',y') \to (-y',x')$
So $Q = (-4, -3)$? No, wait: Wait, 90° clockwise rotation is $(x,y)\to(y,-x)$. So if $Q'=(y,-x)=(-3,4)$, then $y=-3$, $-x=4 \Rightarrow x=-4$. Thus $Q=(x,y)=(-4,-3)$? Wait no—wait the options include (-4,-3) as option 1. Let confirm: If Q is (-4,-3), 90° clockwise rotation: $(y,-x)=(-3,4)$, which is exactly Q'. Yes!

Answer:

(-4, -3) (Option 1)