A triangle has corners at #(3 ,4 )#, #(4 ,7 )#, and #(2 ,8 )#. How far is the triangle's centroid from the origin?

1 Answer
Apr 27, 2016

≈ 7 units

Explanation:

The first step here is to find the coordinates of the centroid
(# x_c" and " y_c)#

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_c = 1/3(x_1 + x_2 + x_3) " and " #

# y_c = 1/3(y_1 + y_2 + y_3)#

here

#x_c = 1/3(3+4+2) = 3" and " y_c=1/3(4+7+8)= 19/3#

coordinates of centroid #= (3 , 19/3)#

To calculate the distance the centroid is from the origin use the #color(blue)" distance formula " #

#color(red)(|bar(ul(color(white)(a/a)color(black)( d =sqrt((x_2 - x_1)^2 + (y_2 -y_1)^2))|)))#
where # (x_1,y_1)" and " (x_2,y_2)" are 2 points "#

The 2 points here are the centroid and the origin
Since the origin (0,0) is one of the points this simplifies the distance formula to

# d =sqrt(3^2 + (19/3)^2)=sqrt(9+361/9) ≈ 7 "units"#