# Thread: Finding the Centroid of this triangle - stuck!

1. ## Finding the Centroid of this triangle - stuck!

I have a triangle CDE

The coordinates for the vertices are C(-2,4) D(6,2) and E(-4,-2)

I have found the mid points of CD (2,3) and CE (-3,1)

I have also found the equation of the median for FE: Y = 5x/6 + 4/3 and GD: Y = x/9 + 4/3

The problem is that I can't solve the system of equations:

1. y= 5x/6 + 4/3
2. y= x/9 + 4/3

I've tried substituting 1. in to 2. but I end up with:
x/9 + 4/3 = 5x/6 + 4/3

Does anyone know where I've gone wrong?

2. ## Re: Finding the Centroid of this triangle - stuck!

If you want to solve the sytem, substitution is a good solution, you'll get:
$\frac{x}{9}+\frac{4}{3}=\frac{5x}{6}+\frac{4}{3}$
$\Leftrightarrow \frac{x}{9}-\frac{5x}{6}=\frac{4}{3}-\frac{4}{3}$
$\Leftrightarrow \frac{x}{9}-\frac{5x}{6}=0$

Get a common denominator and a fraction becomes zero if the numerator because zero ...

Notice if the three vertices of a triangle ABC are $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ then the coordinates of the centroid of the triangle is:
$\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3} \right)$

3. ## Re: Finding the Centroid of this triangle - stuck!

You should know also that:

In triangle $\text{ABC}$, with \text{A}=(x_1,y_1), \text{B}=(x_2,y_2), \text{C}=(x_3,y_3).

The centroid $\text{M}$ is given by:

$\text{M}=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2 +y_3}{3}\right)$.

Try to prove the above using the same arguments that you use with your problem(just replace the coordinates)

Edit:

Sorry @Siron, didn't see your edited post.

4. ## Re: Finding the Centroid of this triangle - stuck!

Thank you for the replies. I came back here with the problem solved however.

I realized that I could multiply each number by 18 since 18 is a common denominator of \frac{x}{9}+\frac{4}{3}=\frac{5x}{6}+\frac{4}{3}

When I did this, I ended up with the correct Centroid. I proved it by making a model in Geometers Sketch Pad and I ended up with the same centroid as the model.

Thanks!

5. ## Re: Finding the Centroid of this triangle - stuck!

You're welcome!

@ Also sprach Zarathrusta:
No problem