# A triangle has corners at #(6 ,9 )#, #(5 ,7 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?

The distance of centroid from origin(0,0) is

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To find the centroid of a triangle with vertices at points ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)), you can use the following formulas:

Centroid coordinates: [ x_c = \frac{x_1 + x_2 + x_3}{3} ] [ y_c = \frac{y_1 + y_2 + y_3}{3} ]

Given the vertices of the triangle as (6, 9), (5, 7), and (3, 8), plug these coordinates into the centroid formulas:

[ x_c = \frac{6 + 5 + 3}{3} = \frac{14}{3} ] [ y_c = \frac{9 + 7 + 8}{3} = \frac{24}{3} = 8 ]

So, the centroid of the triangle is at the point ((\frac{14}{3}, 8)).

To find the distance from the centroid to the origin, you can use the distance formula:

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

Plug in the coordinates of the centroid:

[ d = \sqrt{(\frac{14}{3})^2 + 8^2} = \sqrt{\frac{196}{9} + 64} = \sqrt{\frac{196 + 576}{9}} = \sqrt{\frac{772}{9}} = \frac{\sqrt{772}}{3} ]

So, the distance from the centroid to the origin is ( \frac{\sqrt{772}}{3} ) units.

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To find the centroid of a triangle, you can calculate the average of the x-coordinates and the average of the y-coordinates of its vertices. Then, the coordinates of the centroid (G) can be determined by these averages. After obtaining the centroid's coordinates, you can use the distance formula to find the distance between the centroid and the origin (0,0).

First, find the averages of the x and y coordinates: Average of x-coordinates = (6 + 5 + 3) / 3 = 14 / 3 Average of y-coordinates = (9 + 7 + 8) / 3 = 24 / 3

So, the coordinates of the centroid (G) are (14/3, 8).

Now, calculate the distance between the centroid (14/3, 8) and the origin (0,0) using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2) Distance = √((0 - 14/3)^2 + (0 - 8)^2) Distance = √((196/9) + 64) Distance ≈ √(196/9 + 576/9) Distance ≈ √(772/9) Distance ≈ √(772) / 3 Distance ≈ 27.79 / 3 Distance ≈ 9.26

So, the distance between the centroid of the triangle and the origin is approximately 9.26 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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