A triangle has corners at #(9 ,3 )#, #(7 ,4 )#, and #(3 ,1 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(9  ,3  )#, #(7  ,4  )#, and #(3   ,1  )#.  How far is the triangle's centroid from the origin?

Jayden Jackson · Accepted Answer

Triangle's centroid is #6.872# units away from the origin.

Centroid of a triangle, whose corners are #(x_1,y_1)#, #(x_2,y_2)# and #(x_3,y_3)#, is given by #(1/3(x_1+x_2+x_3),1/3(y_1+y_2+y_3))# Hence centroid of the triangle whose corners are #(9,3)#, #(7.4)# and #(3,1)# is #(1/3(9+7+3),1/3(3+4+1))# or #(19/3,8/3)# And its distance from origin #(0,0)# is #sqrt((19/3-0)^2+(8/3-0)^2)=sqrt(361/9+64/9)# = #1/3sqrt425=1/3xx20.616=6.872#

Mason Adams · Answer

Triangle's centroid from the origin is #6.872# units away.

Centroid of a triangle, whose corners are #(x_1,y_1)#, #(x_2,y_2)# and #(x_3,y_3)#, is given by #(1/3(x_1+x_2+x_3),1/3(y_1+y_2+y_3))# Hence centroid of the triangle whose corners are #(9,3)#, #(7.4)# and #(3,1)# is #(1/3(9+7+3),1/3(3+4+1))# or #(19/3,8/3)# And its distance from origin #(0,0)# is #sqrt((19/3-0)^2+(8/3-0)^2)=sqrt(361/9+64/9)# = #1/3sqrt425=1/3xx20.616=6.872#

Aurora Ayala · Answer

undefined The centroid of a triangle is located at the average of its vertices' coordinates. To find the centroid, add up the coordinates of all vertices and divide by the number of vertices. Then, calculate the distance between the centroid and the origin using the distance formula.

First, find the centroid coordinates:
Centroid_x = (9 + 7 + 3) / 3 = 6.33
Centroid_y = (3 + 4 + 1) / 3 = 2.67

Now, calculate the distance from the origin:
Distance = sqrt((6.33 - 0)^2 + (2.67 - 0)^2)
Distance ≈ sqrt(40.0089 + 7.1289)
Distance ≈ sqrt(47.1378)
Distance ≈ 6.86 units

So, the distance from the origin to the triangle's centroid is approximately 6.86 units.