# A triangle has corners at #(1 ,9 )#, #(7 ,8 )#, and #(4 ,3 )#. How far is the triangle's centroid from the origin?

Distance of centroid from origin

Coordinates of centroid is obtained using the formula

We have to find OG where O is the origin with coordinates (0,0)

Using distance formula,

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

First, find the coordinates of the centroid:

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

Using the given coordinates:

[ \text{Centroid} = \left( \frac{1 + 7 + 4}{3}, \frac{9 + 8 + 3}{3} \right) ] [ \text{Centroid} = \left( \frac{12}{3}, \frac{20}{3} \right) ] [ \text{Centroid} = (4, \frac{20}{3}) ]

Now, use the distance formula to find the distance between the centroid and the origin:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

[ \text{Distance} = \sqrt{(4 - 0)^2 + \left(\frac{20}{3} - 0\right)^2} ] [ \text{Distance} = \sqrt{4^2 + \left(\frac{20}{3}\right)^2} ] [ \text{Distance} = \sqrt{16 + \frac{400}{9}} ] [ \text{Distance} = \sqrt{\frac{144 + 400}{9}} ] [ \text{Distance} = \sqrt{\frac{544}{9}} ] [ \text{Distance} = \frac{\sqrt{544}}{3} ]

Therefore, the distance between the centroid of the triangle and the origin is ( \frac{\sqrt{544}}{3} ) 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 #(9 ,3 )#, #(7 ,5 )#, and #(3 ,4 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(1 ,8 )# and an area of #15 pi#. Circle B has a center at #(5 ,3 )# and an area of #25 pi#. Do the circles overlap?
- 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 #1 #. Do the circles overlap? If not what is the smallest distance between them?
- A triangle has corners at #(3 ,4 )#, #(6 ,3 )#, and #(2 ,8 )#. How far is the triangle's centroid from the origin?
- What is the perimeter of a triangle with corners at #(2 ,6 )#, #(9 ,2 )#, and #(3 ,1 )#?

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