Question

Find the centroid of a triangle whose vertices are $(0,6),(8,12)\text{ and }(8,0)$ .

Answer 1

Hint: To find the value of centroid of a triangle whose vertices are $(0,6),(8,12)\text{ and }(8,0)$ , we will be using the formula \[\text{Centroid }=\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\] . We can now substitute the values from the given data to obtain the centroid of the given vertices.

Complete step by step answer:
We have to find the value of the centroid of a triangle whose vertices are $(0,6),(8,12)\text{ and }(8,0)$ .
Before finding this, let us recollect what a centroid of a triangle is.
Consider a triangle $ABC$ whose vertices are given as \[A({{x}_{1}},{{y}_{1}}),B({{x}_{2,}}{{y}_{2}}),C({{x}_{3}},{{y}_{3}})\] . This is illustrated below:

The centroid of a triangle, denoted as G in the figure, can be calculated by taking the average of X and Y coordinate points of all three vertices. Hence, the centroid of a triangle is given as
Hence, the centroid of a triangle is given as
\[G=\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)...(i)\]
Now, let us compare this with the points given.
We have $(0,6),(8,12)\text{ and }(8,0)$ . Let us rewrite this as shown below.
$A(0,6),B(8,12)\text{ and C}(8,0)$ .
Now, we can compare this with the standard equation. Hence, we get
${{x}_{1}}=0,{{x}_{2}}=8,{{x}_{3}}=8$
\[{{y}_{1}}=6,{{y}_{2}}=12,{{y}_{3}}=0\]
Now we can substitute these in equation (i). We will get
\[G=\left( \dfrac{0+8+8}{3},\dfrac{6+12+0}{3} \right)\]
By solving this, we get
\[G=\left( \dfrac{16}{3},\dfrac{18}{3} \right)\]
Now, let us simplify this further. Hence, the centroid of the given vertices are
\[G=\left( \dfrac{16}{3},6 \right)\]

Thus, the centroid of a triangle whose vertices are $(0,6),(8,12)\text{ and }(8,0)$ is \[\left( \dfrac{16}{3},6 \right)\].

Note: One can make mistake when writing the centroid equation as \[G=\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{2} \right)\] . The easy way to memorize this is that the terms in the numerator of each coordinate is 3. Hence, we will divide the X-coordinate and Y-coordinate by 3.