Determine the median which goes from A in triangle ABC if A(5,2), B(-3,7), C(1,-5).

We'll recall that the median of a triangle is the segment
that joins the vertex of triangle to the midpoint of the opposite
side.

If the median is drawn from vertex A, then the
opposite side is BC.

We'll determine the equation of the
median, and for this reason, we'll determine the midpoint of the segment
[BC].

xM = (xB+xC)/2 => xM = (-3+1)/2 => xM =
-1

yM = (yB+yC)/2 => yM = (7-5)/2 => yM =
1

Since we know two points, we'll write the equation of the
line that passes through the points:

(xM-xA)/(x-xA) =
(yM-yA)/(y-yA)

(-1-5)/(x-5) =
(1-2)/(y-2)

-6/(x-5) =
-1/(y-2)

We'll cross
multiply:

-x + 5 = -6y +
12

We'll shift all the terms to one
side:

-x + 6y + 5 - 12 = 0

x -
6y + 7 = 0

The equation of the median that
joins the vertex A to the midpoint M of the side BC, is x - 6y + 7 =
0.