The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each
side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following.
-(-9, 0), (9, 0), (0, 2)
A) $( 0,2 )$
B) $\left( 0 , \frac { 4 } { 3 } \right)$
C) $( 0,1 )$
D) $\left( 0 , \frac { 2 } { 3 } \right)$

Question

The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each
side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following.
-(-9, 0), (9, 0), (0, 2) 
A) $( 0,2 )$
B) $\left( 0 , \frac { 4 } { 3 } \right)$
C) $( 0,1 )$
D) $\left( 0 , \frac { 2 } { 3 } \right)$

Quizplus · Accepted Answer

We first find the three medians of the triangle:

Median AE passes through (-9,0) and the midpoint of BC, which is $\left( \frac { 9 } { 2 } , 1 \right)$. Therefore, the equation of AE is y=$ \frac{1}{9}(x+9) $.
Median BD passes through (9,0) and the midpoint of AC, which is $\left( -\frac { 9 } { 2 } , 1 \right)$. Therefore, the equation of BD is y=$ -\frac { 1 } { 9 } ( x-9 )$.
Median CF passes through (0,2) and the midpoint of AB, which is (0,0). Therefore, the equation of CF is y=2.

To find the centroid G of the triangle, we need to find the intersection point of the medians. Solving the system of equations, we get:

$ \frac{1}{9}(x+9)=-\frac { 1 } { 9 } ( x-9 ) $

Solving this equation, we get x=0. Therefore, the intersection point of AE and BD is (0,1).

Next we find the intersection of the median CF with the line segment joining A and G. Using the formula for the line segment joining two points $\left( \frac{{x}_{1}+{x}_{2}}{2}, \frac{{y}_{1}+{y}_{2}}{2} \right)$ we find that the midpoint of AG is:
$$\left( \frac { -9 } { 6 } , \frac { 1 } { 2 } \right) = \left( -\frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$$
Therefore, the equation of the line joining C and G is y=2x+2. Substituting x=0, we get y=2. Therefore, the centroid of the triangle is $\left(0, \frac{1}{3}(0+0+2)\right) = \left(0, \frac{2}{3}\right) $. The answer is D.

The Centroid of a Triangle Lies at the Intersection of the Triangle's