# A circle's center is at #(2 ,4 )# and it passes through #(1 ,2 )#. What is the length of an arc covering #(5pi ) /4 # radians on the circle?

Length of the Arc

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To find the length of an arc covering ( \frac{5\pi}{4} ) radians on the circle, we need to first find the radius of the circle. Using the distance formula, we find the distance between the center of the circle (2, 4) and the point on the circle (1, 2). Then, we use the formula for the length of an arc on a circle:

[ \text{Arc Length} = r \times \text{angle in radians} ]

where ( r ) is the radius of the circle.

The distance between the center of the circle (2, 4) and the point on the circle (1, 2) is:

[ \sqrt{(1 - 2)^2 + (2 - 4)^2} = \sqrt{1 + 4} = \sqrt{5} ]

So, the radius of the circle is ( \sqrt{5} ).

Now, we can find the length of the arc:

[ \text{Arc Length} = \sqrt{5} \times \frac{5\pi}{4} = \frac{5\sqrt{5}\pi}{4} ]

Therefore, the length of the arc covering ( \frac{5\pi}{4} ) radians on the circle is ( \frac{5\sqrt{5}\pi}{4} ).

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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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