# A triangle has corners at #(5 ,2 )#, #(4 ,6 )#, and #(3 ,1 )#. How far is the triangle's centroid from the origin?

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To find the centroid of a triangle with vertices at (5, 2), (4, 6), and (3, 1), you would calculate the average of the x-coordinates and the average of the y-coordinates of the vertices. Then, the distance from the centroid to the origin can be found using the distance formula.

First, find the centroid:

(x) coordinate of centroid = ((5 + 4 + 3) / 3 = 4)

(y) coordinate of centroid = ((2 + 6 + 1) / 3 = 3)

So, the centroid is at (4, 3).

Now, calculate the distance from the centroid to the origin:

Distance = (\sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5)

Therefore, the distance from the triangle's centroid to the origin is 5 units.

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To find the centroid of a triangle, you average the coordinates of its vertices.

Centroid coordinates: [ (\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}) ]

Given coordinates: ( (5, 2) ), ( (4, 6) ), ( (3, 1) )

Centroid coordinates: [ (\frac{5 + 4 + 3}{3}, \frac{2 + 6 + 1}{3}) ] [ (\frac{12}{3}, \frac{9}{3}) ] [ (4, 3) ]

Using the distance formula, the distance between the centroid and the origin ( (0, 0) ) is calculated as:

[ d = \sqrt{(x - 0)^2 + (y - 0)^2} ] [ d = \sqrt{(4 - 0)^2 + (3 - 0)^2} ] [ d = \sqrt{4^2 + 3^2} ] [ d = \sqrt{16 + 9} ] [ d = \sqrt{25} ] [ d = 5 ]

So, the distance from the triangle's centroid to the origin is 5 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.

- A triangle has corners at #(5 ,6 )#, #(4 ,7 )#, and #(8 ,9 )#. How far is the triangle's centroid from the origin?
- An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,1 )# to #(2 ,5 )# and the triangle's area is #32 #, what are the possible coordinates of the triangle's third corner?
- What is the perimeter of a triangle with corners at #(9 ,5 )#, #(6 ,3 )#, and #(4 ,7 )#?
- Circle A has a center at #(6 ,2 )# and a radius of #2 #. Circle B has a center at #(5 ,-4 )# and a radius of #3 #. Do the circles overlap? If not what is the smallest distance between them?
- A line passes through #(6 ,2 )# and #(5 ,7 )#. A second line passes through #(3 ,8 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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