**Question:**

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

**Solution:**

We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.

The co-ordinates of other two vertices are (1, 2) and (3, 5)

The co-ordinate of the centroid is (0, 0)

We know that the co-ordinates of the centroid of a triangle whose vertices are $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$ is-

$\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)$

So,

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

Compare individual terms on both the sides-

$\frac{x+1+3}{3}=0$

So,

$x=-4$

Similarly,

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

So,

$y=-7$

So the co-ordinate of third vertex $(-4,-7)$