A triangle has corners at #(2 ,4 )#, #(8 ,6 )#, and #(4 ,9 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(2  ,4  )#, #(8   ,6  )#, and #(4   ,9  )#.  How far is the triangle's centroid from the origin?

Jayden Jackson · Accepted Answer

≈ 7.87 units

Given the vertices of a triangle A#(x_1,y_1),B(x_2,y_2),C(x_3,y_3)# The centroid has coordinates : # x_c = 1/3(x_1 + x_2 + x_3 ) : y_c = 1/3(y_1+y_2+y_3)# For the vertices given : #x_c = 1/3(2+8+4) = 14/3" and " y_c = 1/3(4+6+9) = 19/3# the coordinates of the centroid # = (14/3 , 19/3) # To calculate distance from origin use the #color(blue)" distance formula " # # d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)# since the origin has coordinates (0 , 0) the formula is simplified. #rArr d = sqrt((14/3)^2 + (19/3)^2) # # = sqrt(196/9 + 361/9) = sqrt(557/9) ≈ 7.87#

Jayden Jackson · Answer

undefined To find the centroid of a triangle, you can take the average of the coordinates of its vertices. The centroid is the point where the three medians intersect.

Let's denote the coordinates of the vertices of the triangle as $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $.

The coordinates of the centroid $ (C_x, C_y) $ can be found using the following formulas:

$$ C_x = \frac{x_1 + x_2 + x_3}{3} $$
$$ C_y = \frac{y_1 + y_2 + y_3}{3} $$

Given the coordinates of the vertices:

$$ (x_1, y_1) = (2, 4) $$
$$ (x_2, y_2) = (8, 6) $$
$$ (x_3, y_3) = (4, 9) $$

Substitute these values into the formulas to find $ C_x $ and $ C_y $.

Once you have $ C_x $ and $ C_y $, you can calculate the distance from the centroid to the origin $ (0,0) $ using the distance formula:

$$ d = \sqrt{(C_x - 0)^2 + (C_y - 0)^2} $$