A triangle has corners at #(1 ,9 )#, #(5 ,4 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?

To find the centroid of a triangle, you can find the average of the coordinates of its vertices. The coordinates of the centroid $(x_c, y_c)$ can be calculated using the formula:

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

Where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the vertices.

Using the given coordinates: (1, 9), (5, 4), and (3, 8), we can calculate the centroid.

$x_c = \frac{1 + 5 + 3}{3} = 3$ $y_c = \frac{9 + 4 + 8}{3} = 7$

The coordinates of the centroid are (3, 7).

To find the distance between the centroid and the origin (0, 0), we can use the distance formula:

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

$d = \sqrt{(3 - 0)^2 + (7 - 0)^2}$ $d = \sqrt{3^2 + 7^2}$ $d = \sqrt{9 + 49}$ $d = \sqrt{58}$

So, the distance between the centroid of the triangle and the origin is $\sqrt{58}$ units.