Question

reflect the figure over the line y = -\frac{1}{3}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

For a point $(x_0,y_0)$ reflected over the line $y = mx + c$ (here $m=-\frac{1}{3}, c = - 1$), the general - formula for reflection involves finding the perpendicular distance from the point to the line and then using the mid - point and slope relationships. First, find the equation of the line perpendicular to $y=-\frac{1}{3}x - 1$. The slope of the perpendicular line is $m'=3$ (since the product of slopes of two perpendicular lines is $- 1$).

Step2: Let the point on the figure be $(x_0,y_0)$

The equation of the line perpendicular to $y =-\frac{1}{3}x - 1$ passing through $(x_0,y_0)$ is $y - y_0=3(x - x_0)$, or $y=3x+(y_0 - 3x_0)$.

Step3: Find the intersection point of the two lines

Set $-\frac{1}{3}x - 1=3x+(y_0 - 3x_0)$. Solve for $x$:
\[

$$\begin{align*} -\frac{1}{3}x-1&=3x+(y_0 - 3x_0)\\ -\frac{1}{3}x-3x&=y_0 - 3x_0 + 1\\ -\frac{1 + 9}{3}x&=y_0 - 3x_0+1\\ x&=-\frac{3}{10}(y_0 - 3x_0 + 1) \end{align*}$$

\]
Substitute $x$ into $y =-\frac{1}{3}x - 1$ to get $y$.
Let the intersection point of the two lines be $(x_1,y_1)$.

Step4: Use the mid - point formula

If the original point is $(x_0,y_0)$ and the reflected point is $(x_2,y_2)$, then the mid - point of the line segment joining them $(\frac{x_0 + x_2}{2},\frac{y_0 + y_2}{2})$ is the intersection point $(x_1,y_1)$. So $x_1=\frac{x_0 + x_2}{2}$ and $y_1=\frac{y_0 + y_2}{2}$. We can solve for $x_2$ and $y_2$: $x_2 = 2x_1 - x_0$ and $y_2=2y_1 - y_0$.
Repeat this process for each vertex of the given polygon to find the vertices of the reflected polygon.

Since the vertices of the original polygon are not given numerically, we cannot provide the exact numerical coordinates of the reflected points. But the general procedure to find the reflected figure over the line $y =-\frac{1}{3}x - 1$ is as above.

If we assume a vertex of the original polygon is $(x_0,y_0)$:

1. The equation of the perpendicular line is $y - y_0=3(x - x_0)$.
2. Solve the system of equations

$$\begin{cases}y=-\frac{1}{3}x - 1\\y = 3x+(y_0 - 3x_0)\end{cases}$$

for the intersection point $(x_1,y_1)$.

1. Use the mid - point formula $x_1=\frac{x_0 + x_2}{2}$ and $y_1=\frac{y_0 + y_2}{2}$ to find the coordinates $(x_2,y_2)$ of the reflected point.

Answer:

The steps to find the reflected figure are as above. To get the actual points, the coordinates of the vertices of the original figure need to be known and the above - mentioned calculations need to be carried out for each vertex.