A triangle has corners at #(6 ,7 )#, #(2 ,1 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?

The quick (shortcut) answer is that it is the mean values

To find the centroid of a triangle with vertices at points ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)), you can use the formula:

[ \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]

For the given triangle with vertices at ((6, 7)), ((2, 1)), and ((5, 8)), the centroid is:

[ \left( \frac{6 + 2 + 5}{3}, \frac{7 + 1 + 8}{3} \right) ] [ = \left( \frac{13}{3}, \frac{16}{3} \right) ]

To find the distance between this centroid and the origin ((0, 0)), you use the distance formula:

[ \text{Distance} = \sqrt{(x - 0)^2 + (y - 0)^2} ]

[ = \sqrt{\left(\frac{13}{3}\right)^2 + \left(\frac{16}{3}\right)^2} ] [ = \sqrt{\frac{169}{9} + \frac{256}{9}} ] [ = \sqrt{\frac{425}{9}} ] [ = \frac{\sqrt{425}}{3} ]

So, the distance from the centroid of the triangle to the origin is ( \frac{\sqrt{425}}{3} ) units.