# A triangle has corners at #(7 ,1 )#, #(8 ,2 )#, and #(5 ,9 )#. How far is the triangle's centroid from the origin?

The distance of centroid from the origin is

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To find the centroid of a triangle, you take the average of the coordinates of its vertices. The coordinates of the centroid ( (x_c, y_c) ) are found by averaging the x-coordinates and y-coordinates of the vertices separately.

Given the vertices: A(7, 1), B(8, 2), and C(5, 9).

Average of x-coordinates = ( \frac{7 + 8 + 5}{3} = \frac{20}{3} )

Average of y-coordinates = ( \frac{1 + 2 + 9}{3} = \frac{12}{3} = 4 )

So, the centroid of the triangle is ( (\frac{20}{3}, 4) ).

Now, to find the distance between the centroid and the origin, we use the distance formula:

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

[ = \sqrt{\left(\frac{20}{3}\right)^2 + 4^2} ]

[ = \sqrt{\frac{400}{9} + 16} ]

[ = \sqrt{\frac{400 + 144}{9}} ]

[ = \sqrt{\frac{544}{9}} ]

[ = \frac{\sqrt{544}}{3} ]

[ = \frac{8\sqrt{17}}{3} ]

So, the distance between the centroid of the triangle and the origin is ( \frac{8\sqrt{17}}{3} ) units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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