Question

Two vertices of a triangle are $\left(1,2\right)$, $\left(3,5\right)$ and its centroid is at the origin. Find the coordinates of the third vertex.

Answer 1

Step 1: Let $\left(x,y\right)$ represents the third vertex:

The centroid of a triangle is given by the formula:

$\mathrm{Centroid}=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$. .

Here, let $\left(x_1,y_1\right)=\left(1,2\right)$, $\left(x_2,y_2\right)=\left(3,5\right)$, and $\mathrm{Centroid}=\left(0,0\right)$.

Let the coordinates of the third vertex be $\left(x_3,y_3\right)=\left(x,y\right)$.

$\Rightarrow \left(0,0\right)=\left(\frac{1+3+x}{3},\frac{2+5+y}{3}\right)$

Compare the coordinates.

$\frac{1+3+x}{3}=0$ and $\frac{2+5+y}{3}=0$

Step 2: Determine the value of $x$ and $y$.

Solve $\frac{1+3+x}{3}=0$ to obtain the value of $x$.

$\begin{array}{rcl}1+3+x& =& 0\\ & \Rightarrow & x=-4\end{array}$

Solve $\frac{2+5+y}{3}=0$ to obtain the value of $y$.

$\begin{array}{rcl}2+5+y& =& 0\\ & \Rightarrow & y=-7\end{array}$

Thus, the coordinates of the third vertex is $\left(x,y\right)=\left(-4,-7\right)$

Final Answer: Hence, the coordinates of the third vertex are $\left(-4,-7\right)$.