Question

question 13
find an equation for the perpendicular bisector of the line segment whose endpoints are (7, -1) and (-9, 3).

Explanation:

Step1: Find the mid - point of the line segment

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For the points $(7,-1)$ and $(-9,3)$, the mid - point $M$ is $(\frac{7+( - 9)}{2},\frac{-1 + 3}{2})=(\frac{7-9}{2},\frac{2}{2})=(-1,1)$.

Step2: Find the slope of the line segment

The slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$. For the points $(7,-1)$ and $(-9,3)$, the slope $m_1=\frac{3-( - 1)}{-9 - 7}=\frac{3 + 1}{-16}=-\frac{1}{4}$.

Step3: Find the slope of the perpendicular bisector

If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the perpendicular bisector be $m_2$. Since $m_1\times m_2=-1$ and $m_1 = -\frac{1}{4}$, then $m_2 = 4$.

Step4: Use the point - slope form to find the equation of the perpendicular bisector

The point - slope form of a line is $y - y_0=m(x - x_0)$, where $(x_0,y_0)$ is a point on the line and $m$ is the slope of the line. Using the point $M(-1,1)$ and $m = 4$, we have $y - 1=4(x+1)$. Expanding this gives $y-1 = 4x+4$, or $y=4x + 5$.

Answer:

$y = 4x+5$