Question

The points (2, 3), (1, 5), (3, -2) are the vertices of a triangle. Find the centroid.

Answer 1

HINT: - The centroid is the centre point of the object. The point in which the three medians of the triangle intersect is known as the centroid of the triangle. It is also defined as the point of intersection of all the three medians. The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The centroid of the triangle separates the median in the ratio of 2: 1.

Complete step-by-step solution -
It can be found by taking the average of x- coordinate points and y-coordinate points of all the vertices of the triangle.
Hence, the formula for calculating the centroid of a triangle is given as follows
\[\left( {{x}_{G}},{{y}_{G}} \right)=\left[ \left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3} \right),\left( \dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right) \right]\]
(Where the vertices of the triangle are \[\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right)\ and\ \left( {{x}_{3}},{{y}_{3}} \right)\] )

As mentioned in the question, we have to find the coordinates of the centroid of a triangle and for this we will use the formula mentioned in the hint which is as follows
\[\left( {{x}_{G}},{{y}_{G}} \right)=\left[ \left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3} \right),\left( \dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right) \right]\]
Now, as the vertices of the triangle are given as
(2, 3), (1, 5) and (3, -2)
Hence, the centroid of this triangle will be
\[\begin{align}
  & \left( {{x}_{G}},{{y}_{G}} \right)=\left[ \left( \dfrac{2+1+3}{3} \right),\left( \dfrac{3+5+-2}{3} \right) \right] \\
 & \left( {{x}_{G}},{{y}_{G}} \right)=\left[ \left( \dfrac{6}{3} \right),\left( \dfrac{6}{3} \right) \right] \\
 & \left( {{x}_{G}},{{y}_{G}} \right)=\left( 2,2 \right) \\
\end{align}\]

Hence, the centroid of the triangle is at (2, 2).

NOTE: - The students can make a mistake if they don’t know that the centroid is the centre point of the object. The point in which the three medians of the triangle intersect is known as the centroid of triangle and it can be found by taking the average of x- coordinate points and y-coordinate points of all the vertices of the triangle and this formula can be represented as \[\left( {{x}_{G}},{{y}_{G}} \right)=\left[ \left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3} \right),\left( \dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right) \right]\].