Question

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 for reflecting a point $(x_0,y_0)$ over the line $y=-x + c$ (here $c = 1$) 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 $m = 1$ (since the product of slopes of two - perpendicular lines is - 1). The equation of the perpendicular line passing through $(x_0,y_0)$ is $y - y_0=1\times(x - x_0)$ or $y=x+(y_0 - x_0)$.
Then, find the intersection point of $y=-x + 1$ and $y=x+(y_0 - x_0)$. Set $-x + 1=x+(y_0 - x_0)$. Solving for $x$ gives $2x=x_0 - y_0 + 1$, so $x=\frac{x_0 - y_0+1}{2}$. Substituting $x$ into $y=-x + 1$ gives $y=-\frac{x_0 - y_0+1}{2}+1=\frac{-x_0 + y_0 + 1}{2}$. Let this intersection point be $(x_i,y_i)$.
The reflected point $(x_1,y_1)$ can be found using the mid - point formula. Since the intersection point $(x_i,y_i)$ is the mid - point between $(x_0,y_0)$ and $(x_1,y_1)$, we have $x_i=\frac{x_0 + x_1}{2}$ and $y_i=\frac{y_0 + y_1}{2}$.
The general formula for reflecting a point $(x_0,y_0)$ over the line $y=-x + 1$ is $x_1=1 - y_0$ and $y_1=1 - x_0$.

Step2: Apply formula to each vertex

Let the vertices of the original polygon be $(x_1,y_1),(x_2,y_2),\cdots$. For each vertex $(x_k,y_k)$ of the original figure, calculate the reflected vertex $(x_k',y_k')$ using $x_k'=1 - y_k$ and $y_k'=1 - x_k$. Then plot these new points to get the reflected figure.

Since no specific points of the polygon are given in coordinate form, the general method for reflection over the line $y=-x + 1$ is presented above. If we had the coordinates of the vertices of the polygon (say $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$), we would substitute each $x_i$ and $y_i$ into the formulas $x_i'=1 - y_i$ and $y_i'=1 - x_i$ to get the coordinates of the reflected points $(x_1',y_1'),(x_2',y_2'),\cdots,(x_n',y_n')$ and then plot them.

Answer:

Use the formula $x'=1 - y$ and $y'=1 - x$ to find the coordinates of the reflected points of the vertices of the given polygon and then plot them.