# What is the perpendicular bisector of a line with points at A (-33, 7.5) and B (4,17)?

Mar 11, 2016

Equation of perpendicular bisector is $296 x + 76 y + 3361 = 0$

#### Explanation:

Let us use point slope form of equation, as the desired line passes through mid point of A $\left(- 33 , 7.5\right)$ and B$\left(4 , 17\right)$.

This is given by $\left(\frac{- 33 + 4}{2} , \frac{7.5 + 17}{2}\right)$ or $\left(- \frac{29}{2} , \frac{49}{4}\right)$

The slope of line joining A $\left(- 33 , 7.5\right)$ and B$\left(4 , 17\right)$ is $\frac{17 - 7.5}{4 - \left(- 33\right)}$ or $\frac{9.5}{37}$ or $\frac{19}{74}$.

Hence slope of line perpendicular to this will be $- \frac{74}{19}$, (as product of slopes of two perpendicular lines is $- 1$)

Hence perpendicular bisector will pass through $\left(- \frac{29}{2} , \frac{49}{4}\right)$ and will have a slope of $- \frac{74}{19}$. Its equation will be

$y - \frac{49}{4} = - \frac{74}{19} \left(x + \frac{29}{2}\right)$. To simplify this multiply all by $76$, LCM of the denominators $2 , 4 , 19$. Then this equation becomes

$76 y - \frac{49}{4} \times 76 = - \frac{74}{19} \times 76 \left(x + \frac{29}{2}\right)$ or

$76 y - 931 = - 296 x - 4292$ or $296 x + 76 y + 3361 = 0$