# A triangle has corners at #(1 ,4 )#, #(9 ,6 )#, and #(4 ,5 )#. How far is the triangle's centroid from the origin?

The distance of triangle's centroid from the origin is

and its distance from origin is

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The centroid of a triangle is the point of intersection of its medians. The coordinates of the centroid can be found by taking the average of the coordinates of the vertices. The centroid of a triangle with vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is ((\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})).

For the given triangle with vertices at (1, 4), (9, 6), and (4, 5), the centroid's coordinates are:

[(\frac{1+9+4}{3}, \frac{4+6+5}{3}) = (\frac{14}{3}, \frac{15}{3}) = (\frac{14}{3}, 5)]

The distance from the origin to the centroid is the length of the line segment connecting the origin to the centroid. This distance can be found using the distance formula:

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

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

[d = \sqrt{(\frac{14}{3}-0)^2 + (5-0)^2} = \sqrt{(\frac{14}{3})^2 + 5^2} = \sqrt{\frac{196}{9} + 25} = \sqrt{\frac{196+225}{9}} = \sqrt{\frac{421}{9}} = \frac{\sqrt{421}}{3}]

So, the distance from the centroid to the origin is (\frac{\sqrt{421}}{3}) units.

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

Average the x-coordinates and y-coordinates of the vertices: [ \text{Average of } x \text{-coordinates} = \frac{1 + 9 + 4}{3} = \frac{14}{3} ] [ \text{Average of } y \text{-coordinates} = \frac{4 + 6 + 5}{3} = \frac{15}{3} = 5 ]

So, the centroid is at ( (\frac{14}{3}, 5) ).

Now, calculate the distance from this centroid to the origin: [ \text{Distance} = \sqrt{(\frac{14}{3})^2 + 5^2} ] [ = \sqrt{\frac{196}{9} + 25} ] [ = \sqrt{\frac{196}{9} + \frac{225}{9}} ] [ = \sqrt{\frac{421}{9}} ]

Therefore, the distance from the triangle's centroid to the origin is ( \frac{\sqrt{421}}{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.

- Circle A has a center at #(3 ,2 )# and an area of #13 pi#. Circle B has a center at #(9 ,6 )# and an area of #28 pi#. Do the circles overlap?
- Circle A has a center at #(7 ,-2 )# and a radius of #2 #. Circle B has a center at #(4 ,2 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(11 ,5 )# and an area of #100 pi#. Circle B has a center at #(4 ,9 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?
- A triangle has corners at #(9 ,3 )#, #(7 ,4 )#, and #(3 ,1 )#. How far is the triangle's centroid from the origin?
- A line passes through #(2 ,8 )# and #( 1, 5 )#. A second line passes through #( 6, 1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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