A triangle has corners at #(1 ,6 )#, #(3 ,2 )#, and #(8 ,9 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(1  ,6  )#, #(3  ,2  )#, and #(8   ,9  )#.  How far is the triangle's centroid from the origin?

Violet Aldrich · Accepted Answer

#\sqrt{433}/3=6.936\ ext{unit# Given that the vertices of a triangle are #(x_1, y_1)\equiv(1, 6)#, #(x_2, y_2)\equiv(3, 2)# & #(x_3, y_3)\equiv(8, 9)# then the coordinates of centroid of triangle are given as #(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})# #\equiv (\frac{1+3+8}{3}, \frac{6+2+9}{3})# #\equiv (4, \frac{17}{3})# hence the distance between the centroid #(4, 17/3)# & the origin #(0, 0)# is given by distance formula as follows #\sqrt{(4-0)^2+(17/3-0)^2}# #=\sqrt{433}/3# #=6.936\ ext{unit#

Sadie Adams · Answer

undefined To find the centroid of a triangle, you can average the x-coordinates of the vertices to find the x-coordinate of the centroid, and similarly average the y-coordinates to find the y-coordinate of the centroid.

Centroid's x-coordinate = (1 + 3 + 8) / 3 = 4
Centroid's y-coordinate = (6 + 2 + 9) / 3 = 5.67 (rounded to two decimal places)

Using the distance formula, we can find the distance between the centroid and the origin:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Distance = sqrt((4 - 0)^2 + (5.67 - 0)^2)
Distance = sqrt(16 + 32.16)
Distance ≈ sqrt(48.16)
Distance ≈ 6.93

So, the distance between the triangle's centroid and the origin is approximately 6.93 units.