# A triangle has corners at #(5 ,1 )#, #(7 ,2 )#, and #(6 ,7 )#. How far is the triangle's centroid from the origin?

The distance between the centroid and the origin is

To find the distance from the origin, use the distance formula.

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To find the centroid of a triangle, you take the average of the x-coordinates and the average of the y-coordinates of its vertices. Then, to find the distance between the centroid and the origin, you use the distance formula, considering the centroid's coordinates as one point and the origin as the other point.

Centroid coordinates:
(x_{\text{centroid}} = \frac{5 + 7 + 6}{3} = 6)

(y_{\text{centroid}} = \frac{1 + 2 + 7}{3} = \frac{10}{3})

Distance from the centroid to the origin:
(d = \sqrt{(x_{\text{centroid}} - 0)^2 + (y_{\text{centroid}} - 0)^2})

(d = \sqrt{(6 - 0)^2 + (\frac{10}{3} - 0)^2})

(d = \sqrt{6^2 + (\frac{10}{3})^2})

(d = \sqrt{36 + \frac{100}{9}})

(d = \sqrt{\frac{324}{9} + \frac{100}{9}})

(d = \sqrt{\frac{424}{9}})

(d = \frac{\sqrt{424}}{3})

(d \approx \frac{20.59}{3})

(d \approx 6.86)

So, the distance between the centroid of the triangle and the origin is approximately 6.86 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 line passes through #(4 ,3 )# and #(2 ,6 )#. A second line passes through #(1 ,8 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A triangle has corners at #(4 ,3 )#, #(8 ,6 )#, and #(7 ,1 )#. How far is the triangle's centroid from the origin?
- A line passes through #(2 ,9 )# and #(5 ,6 )#. 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?
- Circle A has a center at #(-2 ,-1 )# and a radius of #3 #. Circle B has a center at #(-8 ,3 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?
- Circle A has a center at #(3 ,4 )# and an area of #16 pi#. Circle B has a center at #(8 ,1 )# and an area of #40 pi#. Do the circles overlap?

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