Centroid

If $\left(-2, 3\right), \left(4, -3 \right) \text{and} \left(4, 5\right)$ are midpoints of the sides of a triangle, find the coordinates of its centroid.

$A(3, 2)$ and $B(-2, 1)$ are two vertices of triangle $ABC$ whose centroid $G$ has coodinates $\left(\frac{5}{3},\frac{-1}{3}\right)$. Find the coordinates of the third vertex $C$ of the triangle.

Two vertices of a triangle are $(3,-5)$ and $(-7, 4)$. If its centroid is $(2,-1)$, find the coordinates of the third vertex.

If the coordinates of the centroid of the triangle whose vertices are $\left(8, 0\right), \left(0, 6\right) \text{and} \left(8, 12\right)$ is $\left(x, y\right)$ then the value of $xy$ is

If $G(-2, 1)$ is the centroid of a triangle $ABC$ and two of its vertices are $A(1,-6) \text{and} B(-5, 2)$, find the third vertex $C$ of the triangle.

If the centroid of the triangle formed by the points $A\left(a, b\right), B\left(b, c\right) \text{and} C\left(c, a\right)$ is at the origin, then the value of $a+b+c$ is?

If the centroid of a triangle $ABC$ whose vertices are $A(2, 2), B(-4, -4) \text{and} C(5, -8)$ is $\left(1, k\right)$, then find the value of $k$

The vertices of triangle $ABC$ are $A(x,y)$, $B(-3, 1)$ and $C(0,-2)$. If its centroid is at the origin, then the value of $\frac{y}{x}$ is

If the coordinates of the centroid of the triangle whose vertices are $(-2, 3), (2, -1), \text{and} (4, 0)$ is $\left(k, \frac{2}{3}\right)$, then find the value of $k$ (as an improper fraction).

If the coordinates of the centroid of the triangle whose vertices are $\left(-3, 0\right), \left(5,-2\right) \text{and} \left(-8, 5\right)$ is $\left(k,1\right)$. Find the value of $k$

If the centroid of the triangle formed by the points $\left(a, b\right), \left(b, c\right) \text{and} \left(c, a\right)$ is at the origin, then $a^3+b^3+c^3=$

If $\left(-2, 1\right)$ is the centroid of the triangle having its vertices at $\left(x, 0\right), \left(5, -2\right), \left(-8, y\right)$, then $x, y$ satisfy the relation

Centroid of a triangle is at origin, then the value of sum of coordinates of all vertices is _____. (write in words)

If $x-2y+k=0$ is a median of the triangle whose vertices are $A\left(-1, 3\right), B\left(0, 4\right) \text{and} C\left(-5, 2\right)$, find the value of $k$.

If the centroid of the triangle formed by $(7, x), (y, -6) \text{and} (9, 10)$ is at $(6, 3)$, then $(x, y)=$

If the centroid of a triangle formed by the points $\left(3, -5\right), \left(-7, 4\right) \text{and} \left(10, -k\right)$ is at the point $\left(k,-1\right)$, then the value of $k$ is

Two vertices of a triangle $ABC$ are $A(-1, 4)$ and $B(5, 2)$ and its centroid is $G(0, -3)$. Then the coordinates of $C$ are

If $A\left(-1,0\right),$ $B\left(5,-2\right)$ and $C\left(8,2\right)$ are the vertices of a triangle $ABC$, then find the coordinates of its centroid.

If $G$ is the centroid of triangle $ABC$ and $P$ any other point, prove that $P{A}^2+P{B}^2+P{C}^2=G{A}^2+G{B}^2+G{C}^2+3G{P}^2$.

If $G$ is the centroid of triangle $ABC$, prove that $B{C}^2+C{A}^2+A{B}^2=3\left(G{A}^2+G{B}^2+G{C}^2\right)$.