A triangle has corners at #(3 ,4 )#, #(6 ,3 )#, and #(2 ,7 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(3  ,4  )#, #(6  ,3  )#, and #(2   ,7  )#.  How far is the triangle's centroid from the origin?

Sadie Adams · Accepted Answer

≈ 5.935 units

The first step is to find the coordinates of the centroid. If #(x_1,y_1),(x_2,y_2)" and " (x_3,y_3) # are the coordinates of the vertices of a triangle , then x-coord#(x_c) = 1/3(x_1+x_2+x_3)" and " # y-coord# (y_c) = 1/3(y_1+y_2+y_3) # here #x_c = 1/3(3+6+2) = 11/3" and " y_c = 1/3(4+3+7) = 14/3 # coords of centroid # = (11/3 , 14/3 ) # To calculate the distance the centroid is from the origin use the #color(blue)" distance formula " # #d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2 )# where #(x_1,y_1)" and " (x_2,y_2)" are 2 coord points "# The 2 points here are the centroid and the origin. Since the origin is one of the points this simplifies the distance formula to : #d = sqrt((11/3)^2 + (14/3)^2) = sqrt((121/9)+(196/9))# # = sqrt(317/9) ≈ 5.935" units " #

Caroline Aucoin · Answer

undefined To find the centroid of a triangle, you can calculate the average of the x-coordinates and the average of the y-coordinates of its vertices. The formula for the centroid (Cx, Cy) of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:

Cx = (x1 + x2 + x3) / 3
Cy = (y1 + y2 + y3) / 3

Using the given coordinates:
x1 = 3, y1 = 4
x2 = 6, y2 = 3
x3 = 2, y3 = 7

Cx = (3 + 6 + 2) / 3 = 11 / 3 ≈ 3.67
Cy = (4 + 3 + 7) / 3 = 14 / 3 ≈ 4.67

So, the centroid of the triangle is approximately (3.67, 4.67).

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

Distance = √((Cx - 0)^2 + (Cy - 0)^2)
         = √((3.67 - 0)^2 + (4.67 - 0)^2)
         ≈ √(13.5169 + 21.8089)
         ≈ √35.3258
         ≈ 5.94

Therefore, the distance between the centroid of the triangle and the origin is approximately 5.94 units.