# What is the equation of the line joining the midpoints of the lines joining (6,2) and (8,4) and (2, 8) and (4, 6)?

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We have to first find the midpoint of the lines segments joining the two pairs of points.

The midpoint of the line segment joining (6,2) and (8,4) is [ (6+8)/2, (2+4)/2] or ( 7, 3). The midpoint of the line segment joining (2,8) and (4,6) is [ (2+4)/2, (8+6)/2] or ( 3, 7).

Now for two points (x1, y1) and (x2, y2) the equation of the line joining them is y-y1 = [(y2-y1)/(x2 – x1)]*(x-x1)

The equation for the line between (3, 7) and (7, 3) is y – 7 = [(3-7)/ (7-3)]*(x-3)

=> y - 7 = (-4/4) (x-3)

=> y - 7 = 3 - x

=> x + y - 10 = 0

**The required equation of the line is x + y - 10 =0**

The mid points of (x1 , y1 ) and (x2, y2) is given by:

M(x , y) = (( x1+x2)/2 , (y1+y2)/2)).

Therefore the coordinates of mid point of (6,2) and (8,4) are ((6+8)/2 , (2+4)/2) = (7, 3).

The coordinates mid point of (2,8) and (4,6) are given by: ((2+4)/2 , (8+6)/2)) = (3,7).

The line joining (7,3) and (3,7) is to be found. We know that the line joining the points (a1 , b1) and (a2, b2) is given by:

y- b1 = {(b2-b1)/(a2-a1)}(x-b1).

Therefore the line joing (7,3) and (3,7) is given by:

y-3 = {(7-3)/(3-7) }(x-7)

y-3 = -1(x-7)

x+y -3-7 = 0

x+y -10 = 0 is the required equation of the line .

The equation of a line that passes through 2 points, whose coordinates are known is:

(xM-xN)/(x-xM) = (yM-yN)/(y-yN)

We'll calculate xM and yM:

xM = (6+8)/2

xM = 7

yM = (2+4)/2

yM = 3

The coordinates of the midpoint of the line that passes through (6,2) and (8,4) is M(7,3).

xN = (2+4)/2

xN = 3

yN = 7

The coordinates of the midpoint of the line that passes through (2, 8) and (4, 6) is N(3,7).

The line that passes through the points M and N is:

(3-7)/(x-7) = (7-3)/(y-3)

-4/(x-7) = 4/(y-3)

We'll divide by 4 both sides:

-1/(x-7) = 1/(y-3)

-y+3 = x-7

**The equation of the line that passes through the midpoints M and N is:**

**x + y - 10 = 0**