# The centroid of a triangle with coordinates

• Mar 18th 2011, 12:12 AM
darksoulzero
The centroid of a triangle with coordinates
Triangle DEF has vertices D(1,3) and E(6,1), and centroid at C(3,4). Determine the coordinates of point F.

I know that the medians in a triangle intersect at the centroid and that the centroid divides each median in a ration of 2:1. Also, I know that each median intersects a side at its midpoint with the shorter part of the median. What I did was find the midpoint between point E and F by taking half of the line D to C and adding onto C to give point G. The coordinates of point G was found to be (4, 4.5). Since I knew that the change in the coordinates between points E and G, (-2, 3.5), was only half of the line EF I doubled it and added it point E to give the answer (2, 8). I wanted to know if there was an easier way to do this.
• Mar 18th 2011, 02:48 AM
mr fantastic
Quote:

Originally Posted by darksoulzero
Triangle DEF has vertices D(1,3) and E(6,1), and centroid at C(3,4). Determine the coordinates of point F.

I know that the medians in a triangle intersect at the centroid and that the centroid divides each median in a ration of 2:1. Also, I know that each median intersects a side at its midpoint with the shorter part of the median. What I did was find the midpoint between point E and F by taking half of the line D to C and adding onto C to give point G. The coordinates of point G was found to be (4, 4.5). Since I knew that the change in the coordinates between points E and G, (-2, 3.5), was only half of the line EF I doubled it and added it point E to give the answer (2, 8). I wanted to know if there was an easier way to do this.

• Mar 26th 2011, 11:16 AM
bjhopper
centroid of triangle
Hi darksoulzero,

Your solution is confusing.Here is a suggested method.

connect M midpoint of DE and C (centroid) with extended lenght.F lies on this line. FC = 2CM. Slope diagram of C and M = 2/1/2.Slope diagram of FC is twice that or 4/1. Working from point C one point left and 4 points up gives F (2,8)

bjh
• Mar 26th 2011, 11:29 AM
mathfun
If ABC is a triangle with $\displaystyle $$A\left( {{x_1},{y_1}} \right),B\left( {{x_2},{y_2}} \right),C\left( {{x_3},{y_3}} \right)$$$ then its centroid is
$\displaystyle $$G\left( {\tfrac{{{x_1} + {x_2} + {x_3}}}{3},\tfrac{{{y_1} + {y_2} + {y_3}}}{3}} \right)$$$
• Mar 26th 2011, 04:52 PM
Soroban
Hello, darksoulzero!

Quote:

$\displaystyle \text{Triangle }D{E}F\text{ has vertices: }D(1,3)\text{ and }E(6,1)\text{, and centroid at }C(3,4).$
$\displaystyle \text{Determine the coordinates of point }F.$
Code:

         1     F o-----+       \    :         \  :4         \  :           \ :           \:0.5       (3,4)o---+     D      C\  :     o        \ :2   (1,3)  *    \:               M o             (3.5,2)  *    E                             o                           (6,1)

We have vertices $\displaystyle D(1,3)$ and $\displaystyle E(6,1)$, and centroid $\displaystyle C(3,4).$

The midpoint of $\displaystyle DE$ is: $\displaystyle M(3\tfrac{1}{2},\,2).$

The median to side $\displaystyle DE$ starts at $\displaystyle \,M$, passes through $\displaystyle \,C,$
. . and extends to $\displaystyle \,F$, where: .$\displaystyle FC \,=\,2\!\cdot\!CM.$

Going from $\displaystyle \,M$ to $\displaystyle \,C$, we move up 2 and left $\displaystyle \frac{1}{2}$
Hence, going from $\displaystyle \,C$ to $\displaystyle \,F$, we move up 4 and left 1.

Therefore, we have: .$\displaystyle F(2,8).$