A triangle has corners at #(6 ,2 )#, #(4 ,3 )#, and #(5 ,1 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(6  ,2  )#, #(4  ,3  )#, and #(5   ,1  )#.  How far is the triangle's centroid from the origin?

Jayden Jackson · Accepted Answer

#\sqrt{29}# units, anout #5.385#.

The centroud of a triangle is obtained from averaging the coordinates of the vertices: Vertices -- #(color(blue)(6),color(gold)(2)),(color(blue)(4),color(gold)(3)),(color(blue)(5),color(gold)(1))# Centroid -- #(color(blue)({6+4+5}/3),color(gold)({2+3+1}/3))=(color(blue)(5),color(gold)(2))#. Then the distance from the origin to point #(a,b)# is #\sqrt{a^2+b^2}#, thus: #\sqrt{5^2+2^2}=\sqrt{29}#.

Sadie Adams · Answer

undefined To find the centroid of a triangle, you can take the average of the coordinates of its vertices. Then, to find the distance from the centroid to the origin, you can use the distance formula.

The coordinates of the centroid (G) can be found by averaging the x-coordinates and y-coordinates separately.

Average x-coordinate: (6 + 4 + 5) / 3 = 15 / 3 = 5
Average y-coordinate: (2 + 3 + 1) / 3 = 6 / 3 = 2

So, the centroid G has coordinates (5, 2).

Now, to find the distance from the centroid to the origin:

Distance = √((x_G)^2 + (y_G)^2)
         = √((5)^2 + (2)^2)
         = √(25 + 4)
         = √29

So, the distance from the centroid to the origin is √29.