Question

reflect the figure over the line y = -x - 1. plot all of the points of the reflected figure. you may click a plotted point to delete it.

Explanation:

Step1: Recall reflection formula

The formula to reflect a point $(x_0,y_0)$ over the line $y = -x + c$ (in our case $c=-1$) is given by: First, find the line perpendicular to $y=-x - 1$ passing through $(x_0,y_0)$. The slope of the perpendicular line is $m = 1$. The equation of the perpendicular line is $y - y_0=1\times(x - x_0)$ or $y=x+(y_0 - x_0)$.

Step2: Find intersection point

Solve the system of equations

$$\begin{cases}y=-x - 1\\y=x+(y_0 - x_0)\end{cases}$$

. Add the two equations: $2y=y_0 - x_0-1$, so $y=\frac{y_0 - x_0-1}{2}$. Substitute $y$ into $y=-x - 1$ gives $\frac{y_0 - x_0-1}{2}=-x - 1$, then $x=-\frac{y_0 - x_0 + 1}{2}$. Let the intersection point be $(x_i,y_i)$.

Step3: Use mid - point formula

If the original point is $(x_0,y_0)$ and the reflected point is $(x_1,y_1)$, and the intersection point of the perpendicular line and the line of reflection is $(x_i,y_i)$, then $x_i=\frac{x_0 + x_1}{2}$ and $y_i=\frac{y_0 + y_1}{2}$. After substitution and simplification, the formula to reflect a point $(x_0,y_0)$ over the line $y=-x - 1$ is $(x_1,y_1)=(-y_0 - 1,-x_0 - 1)$.

Step4: Apply to each vertex

Suppose the vertices of the original quadrilateral are $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. Calculate the reflected vertices using the formula $(x',y')=(-y - 1,-x - 1)$ for each vertex. Then plot these new points on the coordinate - plane.

Since the original vertices are not given, in general, if a point $(x,y)$ is given, its reflection over the line $y=-x - 1$ is $(-y - 1,-x - 1)$.

Answer:

For each point $(x,y)$ of the original figure, the reflected point is $(-y - 1,-x - 1)$. Plot these reflected points.