Centroid, Incentre and Circumcentre and Orthocentre of a Triangle - Practice Questions & MCQ

Centroid   

A median is a line joining the mid-points of a side and the opposite vertex of a triangle.

The Centroid of a triangle is the point of intersection of the three medians of the triangle. A centroid divides any median in the ratio 2:1.

The coordinates of the centroid of a triangle (G) whose vertices are A (x₁, y₁), B (x₂, y₂) and C(x₃, y₃), are given by 

$
\left(\frac{\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3}{3}, \frac{\mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3}{3}\right)
$
Note:
If $D\left(a_1, b_1\right), E\left(a_2, b_2\right)$ and $F\left(a_3, b_3\right)$ are the midpoint of $\triangle A B C$, then the centroid of triangle $A B C$ is given by

$
\left(\frac{\mathbf{a}_1+\mathbf{a}_2+\mathbf{a}_3}{3}, \frac{\mathbf{b}_1+\mathbf{b}_2+\mathbf{b}_3}{3}\right)
$
Example
If Origin is the centroid of a triangle $A B C$, and the coordinates of the other two vertices of the triangle are $A(4,-3)$ and $B(-5,2)$, then find the coordinates of the third vertex.
Solution
Let point $C$ be $(\alpha, \beta)$

$
\begin{aligned}
& (0,0)=\left(\frac{\alpha+4-5}{3}, \frac{\beta-3+2}{3}\right) \\
& \Rightarrow \frac{\alpha+4-5}{3}=0 \text { and } \frac{\beta-3+2}{3}=0 \\
& \alpha=1, \quad \text { and } \beta=1
\end{aligned}
$
Coordinates of $C$ are $(1,1)$