Question

homework: segment jm has endpoints j(-5,1) and m(7,-9). an equation of the perpendicular bisector of jm is 1) y - 4 = 5/6(x + 1) 2) y + 4 = 6/5(x - 1) 3) y - 4 = -6/5(x + 1) 4) y + 4 = -5/6(x - 1)

Explanation:

Step1: Find the mid - point of segment JM

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 $J(-5,1)$ and $M(7,-9)$, the mid - point $P$ is $(\frac{-5 + 7}{2},\frac{1+( - 9)}{2})=(1,-4)$.

Step2: Find the slope of segment JM

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 $J(-5,1)$ and $M(7,-9)$, the slope $m_{JM}=\frac{-9 - 1}{7-( - 5)}=\frac{-10}{12}=-\frac{5}{6}$.

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 $m'$. Then $m_{JM}\times m'=-1$. Since $m_{JM}=-\frac{5}{6}$, we have $-\frac{5}{6}m'=-1$, so $m'=\frac{6}{5}$.

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 mid - point $(1,-4)$ as $(x_0,y_0)$ and $m'=\frac{6}{5}$, we get $y+4=\frac{6}{5}(x - 1)$.

Answer:

$y + 4=\frac{6}{5}(x - 1)$