Question

Find the coordinates of the centroid of a triangle whose vertices are (0, 6), (8, 12) and (8, 0).

Answer 1

Hint: When three vertices of a triangle are \[A({{x}_{1,}}{{y}_{1}}),B({{x}_{2}},{{y}_{2}}),C({{x}_{3}},{{y}_{3}})\], then the centroid of the triangle is given by the formula, \[G\left(\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]. Substitute the given value in the formula accordingly.

We are supposed to find the centroid of the triangle; whose vertices are (0, 6), (8, 12) and (8, 0).
First, let us understand what a centroid is:
The centroid of a triangle is one of the points of concurrency of a triangle. It is the point where all the three medians of a triangle intersect. Median is a line segment which is drawn from a vertex to the midpoint of the opposite side.
Important properties of centroid are:
It is always situated inside the triangle like the incenter.
The centroid of a triangle divides each median in a ratio of 2:1 in single words; the centroid will generally be \[\dfrac{2}{3}rd\] of the way along any median.
Now, let us see how to compute the centroid of a triangle.
The centroid, also known as the center of gravity of the triangle can be found by finding the average of x-coordinates and y-coordinates values of all the three vertices of the triangle.
Centroid of a triangle with vertices\[A({{x}_{1,}}{{y}_{1}}),B({{x}_{2}},{{y}_{2}})andC({{x}_{3}},{{y}_{3}})\]is given as:\[\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]
By substituting the given vertices (0, 6), (8, 12) and (8, 0) in the centroid formula, we get:
Centroid \[=\left( \dfrac{0+8+8}{3},\dfrac{6+12+0}{3} \right)\]
Centroid \[=\left( \dfrac{16}{3},\dfrac{18}{3} \right)\]
Centroid \[=\left( \dfrac{16}{3},6 \right)\]
So, the centroid of the given triangle with the vertices (0, 6), (8, 12) and (8, 0) is \[\left( \dfrac{16}{3},6 \right)\].

Note: Alternatively, we can find the median equations from any two vertices of the triangle and solve these median equations to find the intersection point of them. Since,the centroid is nothing but the intersection of median line segments.