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

The centroud of a triangle is obtained from averaging the coordinates of the vertices:

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To find the centroid of a triangle, you can take the average of the coordinates of its vertices. Then, to find the distance from the centroid to the origin, you can use the distance formula.

The coordinates of the centroid (G) can be found by averaging the x-coordinates and y-coordinates separately.

Average x-coordinate: (6 + 4 + 5) / 3 = 15 / 3 = 5 Average y-coordinate: (2 + 3 + 1) / 3 = 6 / 3 = 2

So, the centroid G has coordinates (5, 2).

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

Distance = √((x_G)^2 + (y_G)^2) = √((5)^2 + (2)^2) = √(25 + 4) = √29

So, the distance from the centroid to the origin is √29.

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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 #(1 ,4 )#, #(7 ,5 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(5 ,-2 )# and a radius of #2 #. Circle B has a center at #(1 ,-4 )# and a radius of #1 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(-4 ,-1 )# and a radius of #3 #. Circle B has a center at #(1 ,3 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?
- A line passes through #(5 ,0 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(1 ,5 )# and #(8 ,7 )#. A second line passes through #(3 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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