A triangle has corners at #(1 ,9 )#, #(5 ,7 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(1  ,9  )#, #(5  ,7  )#, and #(3   ,8  )#.  How far is the triangle's centroid from the origin?

Noah Atkinson · Accepted Answer

#sqrt73# units

If, #(x_1, y_1) => (1, 9)# #(x_2, y_2) => (5, 7)# #(x_3, y_3) => (3, 8)# are the vertices of a triangle then, Centroid (G) of a triangle is given by, #G = ((x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3)# So, substituting the above values we get, #G = (3, 8)# Now, to find distance 'd' between #(0, 0)# and #(3, 8)# use distance formula #d = sqrt((8 - 0)^2 + (3 - 0)^2)# #d = sqrt73# units ...............................................

Theodore Abad · Answer

Distance of centroid from origin is #color(red)(8.544)#

Formula to get centroid of a triangle, given the coordinates of three vertices is

#G (x,y) = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3#

#G_x = (1 + 5 + 3) /3 = 3#

#G_y = (9 + 7 + 8) / 3 = 8#

Coordinates of centroid #G(x,y) = (3,8)#

Coordinates of origin #O(x,y) = (0.0)#

Distance of centroid from origin is
#d = sqrt((3-0)^2 + (8-0)^2) = sqrt(3^2 + 8^2) = **8.544**#

Allison Allen · Answer

undefined To find the centroid of a triangle, you take the average of the x-coordinates and the average of the y-coordinates of its vertices. Then, the distance from the centroid to the origin can be calculated using the distance formula.

Given the coordinates of the vertices: (1, 9), (5, 7), and (3, 8).

The centroid's x-coordinate is the average of 1, 5, and 3, which is (1 + 5 + 3) / 3 = 3.
The centroid's y-coordinate is the average of 9, 7, and 8, which is (9 + 7 + 8) / 3 = 8.

So, the centroid of the triangle is at (3, 8).

To find the distance from the centroid to the origin, we use the distance formula:

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

Where (x1, y1) = (0, 0) (coordinates of the origin) and (x2, y2) = (3, 8) (coordinates of the centroid).

So, Distance = sqrt((3 - 0)^2 + (8 - 0)^2) = sqrt(3^2 + 8^2) = sqrt(9 + 64) = sqrt(73).

Therefore, the distance from the centroid to the origin is sqrt(73).