# Question 040c8

Feb 18, 2017

See the Explanation.

#### Explanation:

Use the Slope-Point Form to find the eqn. of reqd. line (which is,

the Perpendicular Bisector of the Line Segment AB ), to get,

$y - 6 = 2 \left\{x - \left(- 1\right)\right\} , i . e . , y = 2 x + 2 + 6 = 2 x + 8.$

Now, try to find out where you have committed mistake.

Feb 18, 2017

$\left(- 1 , 6\right) \text{ and } y = 2 x + 8$

#### Explanation:

The coordinates of the midpoint are the $\textcolor{b l u e}{\text{average}}$ of the x and y coordinates of A and B.

$\Rightarrow M = \left[\frac{1}{2} \left(1 - 3\right) , \frac{1}{2} \left(5 + 7\right)\right] = \left(- 1 , 6\right)$

We require to calculate the slope( m ) of AB using the $\textcolor{b l u e}{\text{gradient formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

The 2 points here are A(1 ,5) and B(-3 ,7)

let $\left({x}_{1} , {y}_{1}\right) = \left(1 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 7\right)$

$\Rightarrow {m}_{A B} = \frac{7 - 5}{- 3 - 1} = \frac{2}{- 4} = - \frac{1}{2}$

The slope of a line perpendicular to AB is $\textcolor{b l u e}{\text{the negative inverse}}$ of the slope of AB.

rArrm_("perp")=-1/(m_(AB)#

$\Rightarrow {m}_{\text{perp}} = - \frac{1}{- \frac{1}{2}} = 2$

The equation of a line in $\textcolor{b l u e}{\text{point-slope form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ are the coordinates of a point on the line}$

$\text{here "m_("perp")=2" and } \left({x}_{1} , {y}_{1}\right) = M \left(- 1 , 6\right)$

$\Rightarrow y - 6 = 2 \left(x + 1\right)$

distributing and simplifying.

$y = 2 x + 2 + 6$

$\Rightarrow y = 2 x + 8 \text{ is equation of perpendicular line}$