Question

directions: graph and label each figure and its image under the rotation about the give
give the coordinates of the image.

1. parallelogram ( a b c d ) with vertices ( a(6,8), b(10,9), c(9,3) ),

and ( d(5,2) ; 90^{circ} ) counterclockwise rotation about point ( d ).

( a^{prime}(quad, quad) )
( b^{prime}(quad, quad) )
( c^{prime}(quad, quad) )
( d^{prime}(quad, quad) )
rectangle ( w x y z ) with vertices ( w(-3,-4), x(0,-5) ),

Explanation:

Step1: Translate points

Subtract the coordinates of the center of rotation \(D(5,2)\) from each point.
For \(A(6,8)\): \((6 - 5,8 - 2)=(1,6)\)
For \(B(10,9)\): \((10 - 5,9 - 2)=(5,7)\)
For \(C(9,3)\): \((9 - 5,3 - 2)=(4,1)\)
For \(D(5,2)\): \((5 - 5,2 - 2)=(0,0)\)

Step2: Apply \(90^{\circ}\) counter - clockwise rotation formula

The formula for a \(90^{\circ}\) counter - clockwise rotation of a point \((x,y)\) is \((-y,x)\)
For \((1,6)\): \((-6,1)\)
For \((5,7)\): \((-7,5)\)
For \((4,1)\): \((-1,4)\)
For \((0,0)\): \((0,0)\)

Step3: Translate back

Add the coordinates of the center of rotation \(D(5,2)\) to each rotated point.
For \((-6,1)\): \((-6 + 5,1+2)=(-1,3)\)
For \((-7,5)\): \((-7 + 5,5 + 2)=(-2,7)\)
For \((-1,4)\): \((-1+5,4 + 2)=(4,6)\)
For \((0,0)\): \((0 + 5,0+2)=(5,2)\)

Answer:

\(A'(-1,3)\)
\(B'(-2,7)\)
\(C'(4,6)\)
\(D'(5,2)\)