Question

1. what is the outcome of rotating

\\( \triangle abc \\),
where
\\( a ( - 2,2 ) , b ( 5,1 ) , \\) and \\( c ( - 3 , - 1 ) \\)
\\( 90 ^ { \circ } \\) ccw with a center of rotation \\( ( 2 , - 2 ) \\). then a reflection over the \\( y \\)-axis?
\\( a ^ { \prime \prime } ( - 2,2 ) , b ^ { \prime \prime } ( 5,1 ) , \\) and \\( c ^ { \prime \prime } ( - 3 , - 1 ) \\)
\\( a ^ { \prime \prime } ( 2 , - 6 ) , b ^ { \prime \prime } ( 1,1 ) , \\) and \\( c ^ { \prime \prime } ( - 1 , - 7 ) \\)
\\( a ^ { \prime \prime } ( 2 , - 6 ) , b ^ { \prime \prime } ( 1,1 ) , \\) and \\( c ^ { \prime \prime } ( - 1 , - 7 ) \\)
\\( a ^ { \prime \prime } ( - 2 , - 6 ) , b ^ { \prime \prime } ( - 1,1 ) , \\) and \\( c ^ { \prime \prime } ( 1 , - 7 ) \\)

Explanation:

Step1: Rotation formula

The formula for a \(90^{\circ}\) counter - clockwise rotation about a point \((h,k)\) is \((x,y)\to(x - h,y - k)\to-(y - k)+h,(x - h)+k\).
For point \(A(-2,2)\):
\(x=-2,y = 2,h = 2,k=-2\)
First, translate: \(x-2=-2 - 2=-4,y + 2=2+2 = 4\)
After rotation: \(x'=-4,y'=-4\)
Translate back: \(x''=-4+2=-2,y''=-4-2=-6\)

For point \(B(5,1)\):
\(x = 5,y = 1,h = 2,k=-2\)
Translate: \(x-2=5 - 2=3,y + 2=1+2 = 3\)
After rotation: \(x'=-3,y'=3\)
Translate back: \(x''=-3+2=-1,y''=3-2 = 1\)

For point \(C(-3,-1)\):
\(x=-3,y=-1,h = 2,k=-2\)
Translate: \(x-2=-3 - 2=-5,y + 2=-1+2 = 1\)
After rotation: \(x'=-1,y'=-5\)
Translate back: \(x''=-1+2=1,y''=-5-2=-7\)

Step2: Reflection formula

The formula for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\)
For the rotated points:
For \(A(-2,-6)\): after reflection \(A''(2,-6)\)
For \(B(-1,1)\): after reflection \(B''(1,1)\)
For \(C(1,-7)\): after reflection \(C''(-1,-7)\)

Answer:

A(2, - 6), B(1, 1), and C(-1, - 7)