A triangle has corners at #(5 ,1 )#, #(9 ,3 )#, and #(2 ,7 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(5  ,1  )#, #(9  ,3  )#, and #(2   ,7  )#.  How far is the triangle's centroid from the origin?

Claire Akin · Accepted Answer

Triangle's centroid from the origin is #6.472# 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 #(5,1)#, #(9.3)# and #(2,7)# is #(1/3(5+9+2),1/3(1+3+7))# or #(16/3,11/3)# And its distance from origin #(0,0)# is #sqrt((16/3-0)^2+(11/3-0)^2)=sqrt(256/9+121/9)# = #1/3sqrt377=1/3xx19.416=6.472#

Aurora Ayala · Answer

undefined To find the centroid of a triangle, we need to calculate the average of the coordinates of its vertices. The centroid of a triangle is the point where the three medians intersect.

The coordinates of the centroid $ G(x_G, y_G) $ are found by averaging the coordinates of the vertices. If the vertices of the triangle are $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $, then the centroid is given by:

$$ x_G = \frac{{x_1 + x_2 + x_3}}{3} $$
$$ y_G = \frac{{y_1 + y_2 + y_3}}{3} $$

Given the coordinates of the vertices of the triangle:

$ (5, 1) $, $ (9, 3) $, and $ (2, 7) $

We can calculate the centroid:

$$ x_G = \frac{{5 + 9 + 2}}{3} = \frac{{16}}{3} $$
$$ y_G = \frac{{1 + 3 + 7}}{3} = \frac{{11}}{3} $$

Now, we have the coordinates of the centroid as $ (\frac{{16}}{3}, \frac{{11}}{3}) $.

To find the distance between the centroid and the origin, we use the distance formula:

$$ d = \sqrt{{(x_G - 0)^2 + (y_G - 0)^2}} $$

$$ d = \sqrt{{\left(\frac{{16}}{3}\right)^2 + \left(\frac{{11}}{3}\right)^2}} $$

$$ d = \sqrt{{\frac{{256}}{9} + \frac{{121}}{9}}} $$

$$ d = \sqrt{{\frac{{377}}{9}}} $$

$$ d \approx \sqrt{41.89} $$

$$ d \approx 6.47 $$

So, the distance between the centroid of the triangle and the origin is approximately 6.47 units.