Question on "equation of a line". Read the question below.A(1,4), B(3,2) and C(7,5) are vertices of a triangle ABC. Find: 1. the coordinates of the...

The centroid of a triangle is the intercepting point of
the medians of triangle.

The median of a triangle joins the
vertex to the midpoint of the opposite side.

Therefore,
we'll have to determine the equations of two medians of triangle and then to solve the
system of these equations. The solution of the system represents the coordinates of the
centroid.

Let AM be the median that joins the vertex A to
the midpoint of BC, that is M.

To calculate the equation of
the line AM, we'll use the formula:

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

We'll find the midpoint
M:

xM = (xB+xC)/2

xM =
(3+7)/2

xM = 5

yM =
(yB+yC)/2

yM = (2+5)/2

yM =
7/2

(7/2-4)/(y-4) =
(5-1)/(x-1)

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

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

y
- 4 = -x/16 + 1/16

y = -x/16 + 1/16 +
4

y = -x/16 + 65/16 (1)

Let BN
be the median that joins the vertex B to the midpoint of AC, that is
N.

To calculate the equation of the line BN, we'll use the
formula:

(yN-yB)/(y-yB) =
(xN-xB)/(x-xB)

We'll find the midpoint
N:

xN = (xA+xC)/2

xN =
(1+7)/2

xN = 4

yN =
(yA+yC)/2

yN = (4+5)/2

yN =
9/2

(9/2-2)/(y-2) =
(4-3)/(x-3)

y - 2 = 5x/2 -
15/2

y = 5x/2 - 15/2 + 2

y =
5x/2 - 11/2 (2)

Now, we'll solve the system formed by
equations of AM and BN.

We'll equate (1) and
(2):

-x/16 + 65/16 = 5x/2 -
11/2

-x/8 + 65/8 = 5x -
11

We'll move the terms in x to the
left:

-x/8 - 5x = -65/8 -
11

-41x/8 = -153/8

x =
153/41

y = 765/82 - 11/2

y =
314/82