# A triangle has corners at #(5 ,1 )#, #(9 ,3 )#, and #(2 ,7 )#. How far is the triangle's centroid from the origin?

Triangle's centroid from the origin is

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To find the centroid of a triangle, we need to calculate the average of the coordinates of its vertices. The centroid of a triangle is the point where the three medians intersect.

The coordinates of the centroid ( G(x_G, y_G) ) are found by averaging the coordinates of the vertices. If the vertices of the triangle are ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ), then the centroid is given by:

[ x_G = \frac{{x_1 + x_2 + x_3}}{3} ] [ y_G = \frac{{y_1 + y_2 + y_3}}{3} ]

Given the coordinates of the vertices of the triangle:

( (5, 1) ), ( (9, 3) ), and ( (2, 7) )

We can calculate the centroid:

[ x_G = \frac{{5 + 9 + 2}}{3} = \frac{{16}}{3} ] [ y_G = \frac{{1 + 3 + 7}}{3} = \frac{{11}}{3} ]

Now, we have the coordinates of the centroid as ( (\frac{{16}}{3}, \frac{{11}}{3}) ).

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

[ d = \sqrt{{(x_G - 0)^2 + (y_G - 0)^2}} ]

[ d = \sqrt{{\left(\frac{{16}}{3}\right)^2 + \left(\frac{{11}}{3}\right)^2}} ]

[ d = \sqrt{{\frac{{256}}{9} + \frac{{121}}{9}}} ]

[ d = \sqrt{{\frac{{377}}{9}}} ]

[ d \approx \sqrt{41.89} ]

[ d \approx 6.47 ]

So, the distance between the centroid of the triangle and the origin is approximately 6.47 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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