Question

1. reflect quadrilateral abcd over the line y = 2 + x, and label the image abcd. rotate quadrilateral abcd counterclockwise 90° around (-2, -3) as the center of rotation. label the image abcd.

Explanation:

Step1: Recall reflection formula

The formula to reflect a point $(x_0,y_0)$ over the line $y = x + b$ (in this case $b = 2$) is given by:
First, find the line perpendicular to $y=x + 2$ passing through $(x_0,y_0)$. The slope of the perpendicular line is $- 1$. The equation of the perpendicular line is $y-y_0=-(x - x_0)$. Then find the intersection point of $y=x + 2$ and $y-y_0=-(x - x_0)$. Solving the system

$$\begin{cases}y=x + 2\\y=-x+(x_0 + y_0)\end{cases}$$

, we add the two equations: $2y=x + 2-x+(x_0 + y_0)=x_0 + y_0+2$, so $y=\frac{x_0 + y_0+2}{2}$, and $x=y - 2=\frac{x_0 + y_0+2}{2}-2=\frac{x_0 + y_0 - 2}{2}$. Let the intersection point be $P$. Then the reflected point $(x_1,y_1)$ of $(x_0,y_0)$ over $y=x + 2$ satisfies that $P$ is the mid - point of the line segment connecting $(x_0,y_0)$ and $(x_1,y_1)$. Using the mid - point formula $\frac{x_0+x_1}{2}=\frac{x_0 + y_0 - 2}{2}$ and $\frac{y_0+y_1}{2}=\frac{x_0 + y_0+2}{2}$, we get $x_1=y_0 - 2$ and $y_1=x_0+2$. Apply this formula to each vertex of quadrilateral $ABCD$ to get $A'B'C'D'$.

Step2: Recall rotation formula

The formula to rotate a point $(x,y)$ counter - clockwise $90^{\circ}$ around the center of rotation $(a,b)$ is $(x',y')=(a-(y - b),b+(x - a))$. Here $a=-2$ and $b = - 3$. For each vertex $(x,y)$ of $A'B'C'D'$, substitute into the rotation formula: $x'=-2-(y + 3)=-y - 5$ and $y'=-3+(x + 2)=x - 1$. Apply this formula to each vertex of $A'B'C'D'$ to get the vertices of $A''B''C''D''$.

Since the coordinates of $A,B,C,D$ are not given, we cannot calculate the exact numerical values of the vertices of $A'B'C'D'$ and $A''B''C''D''$. But the general procedures for reflection and rotation are as above.

Answer:

The general steps for reflection of quadrilateral $ABCD$ over $y = x+2$ and rotation of the reflected quadrilateral $A'B'C'D'$ counter - clockwise $90^{\circ}$ around $(-2,-3)$ are described above. To get the exact coordinates of $A'B'C'D'$ and $A''B''C''D''$, the coordinates of $A,B,C,D$ need to be provided for further numerical calculations.