Question

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

Answer 1

≈ 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"`