# A circle's center is at #(7 ,5 )# and it passes through #(2 ,7 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?

Length of arc

To maintain precision to not convert to decimal at this point. '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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To find the length of an arc covering ( \frac{3\pi}{4} ) radians on the circle, first, we need to find the radius of the circle. We can use the distance formula between two points, which is:

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

Given the center of the circle at ( (7, 5) ) and a point on the circle at ( (2, 7) ), we can find the distance between them, which is the radius of the circle.

[ r = \sqrt{(2 - 7)^2 + (7 - 5)^2} ]

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

[ r = \sqrt{25 + 4} ]

[ r = \sqrt{29} ]

Now, to find the length of the arc covering ( \frac{3\pi}{4} ) radians on the circle, we use the formula for the length of an arc of a circle:

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

[ \text{Arc Length} = \sqrt{29} \times \frac{3\pi}{4} ]

[ \text{Arc Length} = \frac{3\pi\sqrt{29}}{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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- If the circumference of a circle is half the area, what is the radius of the circle?
- What is the equation of the circle with a center at #(2 ,6 )# and a radius of #5 #?
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