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

The distance is

The distance of the centroid from the origin is

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

Centroid coordinates: (x_{centroid} = \frac{(x_1 + x_2 + x_3)}{3}) (y_{centroid} = \frac{(y_1 + y_2 + y_3)}{3})

Given coordinates of the triangle's vertices: (P_1(4, 3)) (P_2(2, 6)) (P_3(1, 5))

Calculate centroid coordinates: (x_{centroid} = \frac{(4 + 2 + 1)}{3} = \frac{7}{3}) (y_{centroid} = \frac{(3 + 6 + 5)}{3} = \frac{14}{3})

Now, calculate the distance from the centroid to the origin: (distance = \sqrt{(x_{centroid})^2 + (y_{centroid})^2}) (distance = \sqrt{(\frac{7}{3})^2 + (\frac{14}{3})^2}) (distance = \sqrt{\frac{49}{9} + \frac{196}{9}}) (distance = \sqrt{\frac{245}{9}}) (distance \approx 5.78)

Therefore, the distance from the triangle's centroid to the origin is approximately 5.78 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.

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