Question

A triangle has corners at `(5 ,2 )`, `(4 ,6 )`, and `(3 ,5 )`. How far is the triangle's centroid from the origin?

Answer 1

≈ 5.897

Explanation:

The first step is to find the coordinates of the centroid.

If `(x_1,y_1),(x_2,y_2)" and " (x_3,y_3)`
are the coordinates of the vertices of a triangle , then

x-coord `(x_c)=1/3(x_1+x_2+x_3)" and "`

y-coord`(y_c)=1/3(y_1+y_2+y_3)`
so `x_c=1/3(5+4+3)=4" and " y_c=1/3(2+6+5)=13/3`

coords of centroid `=(4,13/3)`

To calculate the distance the centroid is from the origin use the `color(blue)" distance formula "`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 points "`

The 2 points here are (0,0) and `(4,13/3)`

`d=sqrt((4-0)^2+(13/3-0)^2)=sqrt(16+169/9)`

`=sqrt(144/9+169/9)=sqrt(313/9) ≈ 5.897`