A triangle has corners at #(4 ,1 )#, #(6 ,3 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(4  ,1  )#, #(6  ,3  )#, and #(3   ,8  )#.  How far is the triangle's centroid from the origin?

Noah Atkinson · Accepted Answer

Distance of centroid from origin is #5.897# 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 #(4,1)#, #(6,3)# and #(3,8)#, the centroid of given triangle is #((4+6+3)/3,(1+3+8)/3)# or #(13/3,4)#. And its distance from origin is #sqrt((13/3-0)^2+(4-0)^2)=sqrt(169/9+16)=sqrt(169/9+144/9)# = #sqrt((169+144)/9)=sqrt(313/9)=1/3sqrt313=1/3xx17.692=5.897#

Allison Allen · Answer

undefined To find the centroid of a triangle, you calculate the average of the x-coordinates and the average of the y-coordinates of its vertices. The formula for the centroid of a triangle is:

$$ (\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}) $$

Given the coordinates of the triangle's vertices as (4, 1), (6, 3), and (3, 8), we can substitute these values into the formula to find the centroid.

$$ x_{\text{centroid}} = \frac{4 + 6 + 3}{3} = \frac{13}{3} $$
$$ y_{\text{centroid}} = \frac{1 + 3 + 8}{3} = \frac{12}{3} = 4 $$

So, the coordinates of the centroid are $(\frac{13}{3}, 4)$.

To find the distance between the centroid and the origin, we can use the distance formula, which is the Pythagorean theorem in two dimensions:

$$ d = \sqrt{(x_{\text{centroid}} - 0)^2 + (y_{\text{centroid}} - 0)^2} $$

Substitute the coordinates of the centroid into the formula:

$$ d = \sqrt{(\frac{13}{3})^2 + 4^2} $$
$$ d = \sqrt{\frac{169}{9} + 16} $$
$$ d = \sqrt{\frac{169}{9} + \frac{144}{9}} $$
$$ d = \sqrt{\frac{313}{9}} $$
$$ d = \frac{\sqrt{313}}{3} $$

So, the distance between the centroid of the triangle and the origin is $\frac{\sqrt{313}}{3}$.