# A circle's center is at #(9 ,7 )# and it passes through #(6 ,2 )#. What is the length of an arc covering #(5 pi ) /6 # radians on the circle?

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To find the length of an arc covering ( \frac{5\pi}{6} ) radians on the circle, you can use the formula:

[ \text{Arc Length} = r \cdot \theta, ]

where ( r ) is the radius of the circle and ( \theta ) is the angle in radians subtended by the arc at the center of the circle.

First, you need to find the radius of the circle using the given center and a point on the circle.

Given the center (9, 7) and a point (6, 2) on the circle, you can use the distance formula:

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

to find the radius.

[ \text{Distance} = \sqrt{(6 - 9)^2 + (2 - 7)^2} ]

[ \text{Distance} = \sqrt{(-3)^2 + (-5)^2} ]

[ \text{Distance} = \sqrt{9 + 25} ]

[ \text{Distance} = \sqrt{34} ]

So, the radius of the circle is ( \sqrt{34} ).

Now, plug in the values into the arc length formula:

[ \text{Arc Length} = (\sqrt{34}) \cdot \frac{5\pi}{6} ]

[ \text{Arc Length} = \frac{5\pi\sqrt{34}}{6} ]

Therefore, the length of the arc covering ( \frac{5\pi}{6} ) radians on the circle is ( \frac{5\pi\sqrt{34}}{6} ).

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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 circle has a center that falls on the line #y = 2/9x +8 # and passes through # ( 3 ,5 )# and #(1 ,4 )#. What is the equation of the circle?
- A triangle has corners at #(9 ,3 )#, #(3 ,9 )#, and #(5 ,8 )#. What is the area of the triangle's circumscribed circle?
- A triangle has corners at #(2 ,4 )#, #(3 ,6 )#, and #(4 ,7 )#. What is the area of the triangle's circumscribed circle?
- A circle has a chord that goes from #pi/3 # to #pi/2 # radians on the circle. If the area of the circle is #9 pi #, what is the length of the chord?
- A triangle has sides with lengths of 3, 5, and 2. What is the radius of the triangles inscribed circle?

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