# Find the function of the line passes through mid-point of AB where A(2,3) B(0,5) and perpendicular to the line 3y - 6x +3 = 0

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The line passes through the point m which is midpoint of AB

A(2,3) and B(0,5)

To find the point (m) we will use the midpoint formula to determine the coordinates:

We know that:

xm = (xA+ xB)/2 = (2+0)/2 = 1

ym = (yA+yB)/2 = (3+5)/2 = 4

Then the coordinates of the point m is ( 1,4)

Then we will write the equation for the line:

( y- ym) = m(x-m1) where m is the slope

==> ( y-4) = m(x-1)

Now we know that the line is perpendicular to the line : 3y - 6x + 3 = 0

==> Let us rewrtie in the slope form.

==> 3y = 6x - 3

==> y= 2x - 1

Since they are perpendicular , then the product of the slopes = -1

==> 2 * m = -1

==> m = -1/2

==> (y-4) = (-1/2) ( x- 1)

==> y= (-1/2)x + 1/2 + 4

==> y= (-1/2)x +9/2

==> 2y = -x + 9

**==> 2y+x - 9 = 0 **

Mid point of a line segment AB with coordinates A(x1, Y1) and B)x2, y20 is given by:

Mid point = [(x1 + x2)/2 + (y1 + y2)/2]

Substituting given values of coordinates of A and B

Mid point = [(2 + 0)/2 + (3 + 5)/2] = (1, 4)

Slope of a given line with equation ax + by + c = 0 is given by:

Slope = -a/b

Therefore slope of the given line, that is 3y - 6x + 3 = 0 is

Slope = m1 = 6/3 = 2

Slope of line perpendicular to it (m2) is given by:

m2 = -1/m1 = -1/2

Then equation of the perpendicular line is:

y = -x/2 + c

This line passes through e point (1, 4). Substituting these values above in the equation of line:

4 = -1/2 + c

==> c = 4 + 1/2 = 9/2

Substituting this value of c in equation of line:

y = -x/2 + 9/2

Answer:

The line can be represented by the function (f(x) = y such that:

y = -x/2 + 9/2