# A triangle has corners at #(4 ,7 )#, #(2 ,3 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?

The distance is

The centroid of the triangle is

The distance of the centroid frm the origin is

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To find the centroid of a triangle, you average the x-coordinates of the vertices to find the x-coordinate of the centroid, and similarly average the y-coordinates to find the y-coordinate of the centroid. Then, you can use the distance formula to find the distance between the centroid and the origin.

The x-coordinate of the centroid ( \bar{x} ) is the average of the x-coordinates of the vertices, and the y-coordinate of the centroid ( \bar{y} ) is the average of the y-coordinates of the vertices.

[ \bar{x} = \frac{x_1 + x_2 + x_3}{3} ] [ \bar{y} = \frac{y_1 + y_2 + y_3}{3} ]

For the given points: (4, 7), (2, 3), and (5, 8),

[ \bar{x} = \frac{4 + 2 + 5}{3} = \frac{11}{3} ] [ \bar{y} = \frac{7 + 3 + 8}{3} = \frac{18}{3} = 6 ]

The coordinates of the centroid are ( (\frac{11}{3}, 6) ).

Now, using the distance formula, the distance between the centroid and the origin is:

[ d = \sqrt{(\bar{x} - 0)^2 + (\bar{y} - 0)^2} ] [ d = \sqrt{(\frac{11}{3})^2 + 6^2} ] [ d = \sqrt{\frac{121}{9} + 36} ] [ d = \sqrt{\frac{121 + 324}{9}} ] [ d = \sqrt{\frac{445}{9}} ] [ d = \frac{\sqrt{445}}{3} ]

So, the distance between the centroid and the origin is ( \frac{\sqrt{445}}{3} ).

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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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