Question

A triangle has corners at `(4 ,6 )`, `(8 ,3 )`, and `(3 ,5 )`. How far is the triangle's centroid from the origin?

Answer 1

Approximately `6.84` units

Explanation:

In general , given a triangle's corners at:
`color(white)("XXX")P = (P_x,P_y)`
`color(white)("XXX")Q=(Q_x,Q_y)`
`color(white)("XXX")R=(R_x,R_y)`
the centroid is located at
`color(white)("XXX")C= ((P_x+Q_x+r_x)/3,(P_y+Q_y+R_y)/3)`

In this case with corners `(4,6), (8,3), and (3,5)`
the centroid is at
`color(white)("XXX")((4+8+3)/3,(6+3+5)/3)=(5,14/3)`

The distance from the origin (i.e. from `(0,0)`) is given by the Pythagorean Theorem
`color(white)("XXX")d=sqrt((5-0)^2+(14/3)^2)`

`color(white)("XXX")=sqrt(25xx9+196)/3`

`color(white)("XXX")=sqrt(421)/3`

`color(white)("XXX")~~6.84`