# A triangle has corners at #(1 ,9 )#, #(3 ,4 )#, and #(4 ,5 )#. How far is the triangle's centroid from the origin?

≈ 6.566 units.

That is the average of the coordinates of the vertices.

substituting the given coordinates into the above.

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The centroid of a triangle is the point where the three medians intersect. The coordinates of the centroid (G) of a triangle with vertices (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)) are given by:

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

Using the coordinates provided: (A(1, 9)), (B(3, 4)), and (C(4, 5))

[G\left(\frac{1 + 3 + 4}{3}, \frac{9 + 4 + 5}{3}\right)]

[G\left(\frac{8}{3}, \frac{18}{3}\right)]

[G\left(\frac{8}{3}, 6\right)]

Now, using the distance formula, the distance between the centroid (G) and the origin ((0, 0)) is:

[d = \sqrt{(x_G - x_0)^2 + (y_G - y_0)^2}]

[d = \sqrt{\left(\frac{8}{3} - 0\right)^2 + (6 - 0)^2}]

[d = \sqrt{\left(\frac{8}{3}\right)^2 + 6^2}]

[d = \sqrt{\frac{64}{9} + 36}]

[d = \sqrt{\frac{64}{9} + \frac{324}{9}}]

[d = \sqrt{\frac{388}{9}}]

[d \approx \frac{\sqrt{388}}{3}]

Thus, the distance from the centroid of the triangle to the origin is approximately (\frac{\sqrt{388}}{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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