# Question #9cca2

May 2, 2017

$\text{The point that we are looking for is D(4,1).}$

#### Explanation:

$\text{1-draw the points A(1,2) and B(3,4)}$
$\text{2-draw a line segment AB}$
$\text{3-find midpoint of the line segment AB}$
$C \left(x , y\right) = \left(\frac{1 + 3}{2} , \frac{2 + 4}{2}\right) \text{ , } C \left(2 , 3\right)$

$\text{4-draw a line(black and dotted) passing C(2,3) and}$
$\text{perpendicular to the line segment AB}$

$\text{since the slope of the AB is 1,the slope of the black line is -1}$
$\text{we can find out the equation of black dotted line.}$
$y - 3 = - 1 \left(x - 2\right)$
$y = - 1 \left(x - 2\right) + 3$
$y = - x + 2 + 3$
$y = - x + 5$
$x + y = 5$

$\text{now we have two equation.}$

$2 x - 3 y = 5 \text{ } \left(1\right)$
$x + y = 5 \text{ } \left(2\right)$

$\text{let us expand the equation (2) by 3}$

$3 \left(x + y\right) = 3 \cdot 5$
$3 x + 3 y = 15 \text{ } \left(3\right)$

$\text{let us sum (1) and (3)}$

$2 x - \cancel{3 y} + 3 x + \cancel{3 y} = 5 + 15$

$5 x = 20$

$x = \frac{20}{5} = 4$

$\text{now use (1) or (2)}$

$x + y = 5$

$4 + y = 5$

$y = 5 - 4$

$y = 1$