A triangle has corners at #(5 ,6 )#, #(2 ,7 )#, and #(3 ,5 )#. How far is the triangle's centroid from the origin?
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To find the centroid of a triangle with vertices at (5, 6), (2, 7), and (3, 5), follow these steps:

Find the coordinates of the centroid, which is the average of the xcoordinates and the average of the ycoordinates of the vertices.

Average of xcoordinates: [ \text{Average of } x = \frac{5 + 2 + 3}{3} = \frac{10}{3} ]

Average of ycoordinates: [ \text{Average of } y = \frac{6 + 7 + 5}{3} = 6 ]

So, the coordinates of the centroid are ( \left(\frac{10}{3}, 6\right) ).

To find the distance between the centroid and the origin, use the distance formula: [ \text{Distance} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ]

Plug in the coordinates of the centroid ( (\frac{10}{3}, 6) ) and the origin ( (0, 0) ): [ \text{Distance} = \sqrt{\left(\frac{10}{3}  0\right)^2 + (6  0)^2} ]

Simplify: [ \text{Distance} = \sqrt{\left(\frac{10}{3}\right)^2 + 6^2} ] [ = \sqrt{\frac{100}{9} + 36} ] [ = \sqrt{\frac{100}{9} + \frac{324}{9}} ] [ = \sqrt{\frac{424}{9}} ] [ = \frac{\sqrt{424}}{3} ]

Approximate the square root of 424: [ \sqrt{424} \approx 20.59 ]

So, the distance between the centroid and the origin is approximately: [ \frac{20.59}{3} \approx \boxed{6.86} ]
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When evaluating a onesided 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 onesided 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 onesided 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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