# A triangle has corners at #(4 ,1 )#, #(8 ,3 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?

The distance is

The coordinates of the centroid are

and

and the distance from the orogin is

Here, we have

Therefore,

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To find the centroid of a triangle with vertices at (4, 1), (8, 3), and (5, 8), you can use the formula:

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

Substitute the coordinates of the vertices:

[ x_1 = 4, \quad y_1 = 1 ] [ x_2 = 8, \quad y_2 = 3 ] [ x_3 = 5, \quad y_3 = 8 ]

[ \text{Centroid} = \left( \frac{4 + 8 + 5}{3}, \frac{1 + 3 + 8}{3} \right) ] [ \text{Centroid} = \left( \frac{17}{3}, \frac{12}{3} \right) ] [ \text{Centroid} = \left( \frac{17}{3}, 4 \right) ]

The distance from the origin to the centroid can be found using the distance formula:

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

Where ( (x_c, y_c) ) are the coordinates of the centroid.

Substitute the coordinates of the centroid:

[ d = \sqrt{\left(\frac{17}{3}\right)^2 + 4^2} ] [ d = \sqrt{\frac{289}{9} + 16} ] [ d = \sqrt{\frac{289}{9} + \frac{144}{9}} ] [ d = \sqrt{\frac{433}{9}} ]

So, the distance from the origin to the centroid of the triangle is ( \sqrt{\frac{433}{9}} ) 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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- A triangle has corners at #(7 ,1 )#, #(8 ,2 )#, and #(5 ,9 )#. How far is the triangle's centroid from the origin?
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- Circle A has a center at #(3 ,2 )# and a radius of #2 #. Circle B has a center at #(1 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(1 ,4 )# and a radius of #2 #. Circle B has a center at #(9 ,3 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

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