A triangle has corners at #(9 ,5 )#, #(2 ,7 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(9  ,5  )#, #(2  ,7  )#, and #(3   ,2  )#.  How far is the triangle's centroid from the origin?

Allison Allen · Accepted Answer

The distance of centroid from the origin is #6.6# unit.

The co-ordinate #(x,y)# of centroid is #x=(x_1+x_2+x_3)/3 and y=(y_1+y_2+y_3)/3 or x= (9+2+3)/3=14/3 ; y=(5+7+2)/3 =14/3# The distance of centroid #(14/3,14/3)# from the origin #(0,0)# is #d=sqrt((14/3-0)^2 +(14/3-0)^2)=sqrt (196/9+196/9) =6.6 unit#[Ans]

Ella Armstrong · Answer

undefined To find the centroid of a triangle with vertices at points $ A(x_1, y_1) $, $ B(x_2, y_2) $, and $ C(x_3, y_3) $, you can use the following formulas:

$$ x_c = \frac{x_1 + x_2 + x_3}{3} $$
$$ y_c = \frac{y_1 + y_2 + y_3}{3} $$

Given the vertices $ A(9, 5) $, $ B(2, 7) $, and $ C(3, 2) $, you can calculate the coordinates of the centroid.

1. Calculate $ x_c $:
$$ x_c = \frac{9 + 2 + 3}{3} $$
$$ x_c = \frac{14}{3} $$

2. Calculate $ y_c $:
$$ y_c = \frac{5 + 7 + 2}{3} $$
$$ y_c = \frac{14}{3} $$

The coordinates of the centroid $ G(x_c, y_c) $ are $ \left(\frac{14}{3}, \frac{14}{3}\right) $.

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

$$ \text{Distance} = \sqrt{(x_c - 0)^2 + (y_c - 0)^2} $$

Substitute the coordinates of the centroid $ G $ into this formula:

$$ \text{Distance} = \sqrt{\left(\frac{14}{3}\right)^2 + \left(\frac{14}{3}\right)^2} $$

$$ \text{Distance} = \sqrt{\frac{196}{9} + \frac{196}{9}} $$

$$ \text{Distance} = \sqrt{\frac{392}{9}} $$

$$ \text{Distance} = \frac{\sqrt{392}}{3} $$

So, the distance from the centroid of the triangle to the origin is $ \frac{\sqrt{392}}{3} $ units.