Question

chapter ii.5 score: 3/5 answered: 4/5 question 5 take l to be the line that is given by the equation l = -14x + 11. reflect the point (7, -75) across l.

Explanation:

Step1: Find the slope of the perpendicular line

The slope of line $L$ is $m_1=- 14$. The slope of a line perpendicular to $L$, $m_2$, satisfies $m_1m_2=-1$. So $m_2=\frac{1}{14}$.

Step2: Write the equation of the line passing through the given point with the perpendicular - slope

The equation of a line in point - slope form is $y - y_1=m_2(x - x_1)$. Using the point $(x_1,y_1)=(7,-75)$ and $m_2=\frac{1}{14}$, we have $y+75=\frac{1}{14}(x - 7)$, which simplifies to $y=\frac{1}{14}x-\frac{7}{14}-75=\frac{1}{14}x-\frac{1}{2}-75=\frac{1}{14}x-\frac{1 + 150}{2}=\frac{1}{14}x-\frac{151}{2}$.

Step3: Find the intersection point of the two lines

We solve the system of equations

$$\begin{cases}y=-14x + 11\\y=\frac{1}{14}x-\frac{151}{2}\end{cases}$$

. Set $-14x + 11=\frac{1}{14}x-\frac{151}{2}$. Multiply through by 14 to clear the fraction: $-196x+154=x - 1057$. Rearrange terms: $-196x - x=-1057 - 154$, $-197x=-1211$, so $x = \frac{1211}{197}$. Substitute $x=\frac{1211}{197}$ into $y=-14x + 11$: $y=-14\times\frac{1211}{197}+11=\frac{-16954+2167}{197}=\frac{-14787}{197}$. Let the intersection point be $(x_0,y_0)=(\frac{1211}{197},\frac{-14787}{197})$.

Step4: Use the mid - point formula to find the reflected point

Let the reflected point be $(x_2,y_2)$. The mid - point of the line segment connecting $(7,-75)$ and $(x_2,y_2)$ is the intersection point $(\frac{1211}{197},\frac{-14787}{197})$. Using the mid - point formula $\frac{7 + x_2}{2}=\frac{1211}{197}$ and $\frac{-75 + y_2}{2}=\frac{-14787}{197}$.
For $\frac{7 + x_2}{2}=\frac{1211}{197}$, we have $7 + x_2=\frac{2422}{197}$, $x_2=\frac{2422}{197}-7=\frac{2422-1379}{197}=\frac{1043}{197}$.
For $\frac{-75 + y_2}{2}=\frac{-14787}{197}$, we have $-75 + y_2=\frac{-29574}{197}$, $y_2=\frac{-29574}{197}+75=\frac{-29574 + 14775}{197}=\frac{-14799}{197}$.

Answer:

$(\frac{1043}{197},\frac{-14799}{197})$