Question

A triangle has corners at `(6 ,7 )`, `(2 ,6 )`, and `(1 ,2 )`. How far is the triangle's centroid from the origin?

Answer 1

≈ 5.83 units

Explanation:

The first step here is to calculate the coordinates of the centroid.

Given 3 vertices `(x_1,y_1),(x_2,y_2),(x_3,y_3)` then the coordinates of the centroid are found as follows.

x-coordinate `x_c=color(red)(|bar(ul(color(white)(a/a)color(black)(1/3(x_1+x_2+x_3))color(white)(a/a)|)))`

and y-coordinate `y_c=color(red)(|bar(ul(color(white)(a/a)color(black)(1/3(y_1+y_2+y_3))color(white)(a/a)|)))`

Using the 3 vertices given here.

`x_c=1/3(6+2+1)=1/3xx9=3`

and `y_c=1/3(7+6+2)=1/3xx15=5`

Hence coordinates of centroid = (3 ,5)

To calculate the distance between (3 ,5) and (0 ,0) use `color(blue)"Pythagoras' theorem"`

`d=sqrt(3^2+5^2)=sqrt(9+25)=sqrt34≈5.83" units"`