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

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To find the centroid of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3), you can use the formula:

Centroid = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

For the given triangle with vertices at (1, 4), (3, 4), and (6, 2), the coordinates of the centroid can be calculated as follows:

x-coordinate of centroid = (1 + 3 + 6) / 3 = 10 / 3 y-coordinate of centroid = (4 + 4 + 2) / 3 = 10 / 3

So, the centroid of the triangle is at the point (10/3, 10/3).

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

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates of the centroid (10/3, 10/3) and the origin (0, 0) into the formula:

Distance = √((10/3 - 0)^2 + (10/3 - 0)^2) = √((100/9) + (100/9)) = √(200/9) = √(200) / 3 ≈ 6.67

So, the distance between the centroid of the triangle and the origin is approximately 6.67 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.

- An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(4 ,1 )# to #(8 ,5 )# and the triangle's area is #64 #, what are the possible coordinates of the triangle's third corner?
- Circle A has a center at #(1 ,-2 )# and a radius of #2 #. Circle B has a center at #(4 ,3 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
- A triangle has corners at #(5 ,2 )#, #(9 ,7 )#, and #(3 ,5 )#. How far is the triangle's centroid from the origin?
- What is the perimeter of a triangle with corners at #(7 ,3 )#, #(4 ,5 )#, and #(3 ,1 )#?
- A triangle has corners at #(1 ,4 )#, #(3 ,5 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?

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