A triangle has corners at #(2 ,9 )#, #(7 ,8 )#, and #(4 ,3 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(2  ,9  )#, #(7  ,8  )#, and #(4   ,3  )#.  How far is the triangle's centroid from the origin?

Mason Adams · Accepted Answer

The triangle's centroid is #7.951# units from the origin.

Coordinates of centroid of a triangle whose vertices (corners) are #(x_1,y_1)#, #(x_2,y_2)# and #(x_3,y_3)# is given by #((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)# As corners of triangle are #(2,9)#, #(7,8)# and #(4,3)#, the centroid of given triangle is #((2+7+4)/3,(9+8+3)/3)# or #(13/3,20/3)#. And its distance from origin is #sqrt((13/3-0)^2+(20/3-0)^2)=sqrt(169/9+400/9)=sqrt(569/9)# = #1/3sqrt569=1/3xxsqrt74=1/3xx23.854=7.951#

Wyatt Anderson · Answer

undefined To find the centroid of a triangle and its distance from the origin, follow these steps:

1. Find the coordinates of the centroid by averaging the x-coordinates and y-coordinates of the triangle's vertices.
2. Once you have the centroid's coordinates, calculate the distance between the centroid and the origin using the distance formula.

Let's calculate:

1. Find the centroid's coordinates:
   Average of x-coordinates: (2 + 7 + 4) / 3 = 13 / 3
   Average of y-coordinates: (9 + 8 + 3) / 3 = 20 / 3
   So, the centroid's coordinates are (13/3, 20/3).

2. Calculate the distance between the centroid and the origin:
   Distance = √((x2 - x1)^2 + (y2 - y1)^2)
            = √((13/3 - 0)^2 + (20/3 - 0)^2)
            = √((169/9) + (400/9))
            = √(569/9)
            ≈ 7.52 units

Therefore, the distance between the centroid of the triangle and the origin is approximately 7.52 units.