Let $D$ be a region bounded by a simple closed path $C$ in the $x y$. Then the coordinates of the centroid $( \bar { x } , \bar { y } )$ of $D$ are $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where $A$ is the area of $D$.
Find the centroid of the triangle with vertices $( 0,0 ) , ( 5,0 )$ and $( 0,10 )$.
A) $\left( \frac { 1 } { 8 } , \frac { 1 } { 8 } \right)$
B) $\left( \frac { 10 } { 3 } , \frac { 10 } { 3 } \right)$
C) $\left( \frac { 1 } { 10 } , \frac { 1 } { 5 } \right)$
D) $\left( \frac { 5 } { 3 } , - \frac { 10 } { 3 } \right)$
E) $\left( \frac { 1 } { 8 } , \frac { 1 } { 64 } \right)$

Question

Quizplus · Accepted Answer

The centroid of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is given by $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$. For the given triangle, the centroid is $\left(\frac{0 + 5 + 0}{3}, \frac{0 + 0 + 10}{3}\right) = \left(\frac{5}{3}, \frac{10}{3}\right)$.

Let DDD Be a Region Bounded by a Simple Closed Path CCC

Let $D$ Be a Region Bounded by a Simple Closed Path $C$