Points #(6 ,7 )# and #(7 ,5 )# are #(2 pi)/3 # radians apart on a circle. What is the shortest arc length between the points?
Refer to the figure below
Applying Law of Cosines in Length of the arc
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To find the shortest arc length between two points on a circle, you can use the formula for arc length (s) given by:
[s = r \cdot \theta]
Where (r) is the radius of the circle and (\theta) is the central angle between the two points measured in radians.
Given that the points are ((6, 7)) and ((7, 5)) and they are (\frac{2\pi}{3}) radians apart, we first find the radius of the circle using the distance formula:
[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]
Plugging in the coordinates, we get:
[d = \sqrt{(7 - 6)^2 + (5 - 7)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}]
Now, we use the formula for arc length:
[s = r \cdot \theta]
[s = \sqrt{5} \cdot \frac{2\pi}{3} = \frac{2\pi\sqrt{5}}{3}]
So, the shortest arc length between the points is (\frac{2\pi\sqrt{5}}{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.
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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- A circle has a center that falls on the line #y = 5/4x +4 # and passes through # ( 4 ,7 )# and #(2 ,5 )#. What is the equation of the circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #( pi)/4 #, and the triangle's area is #48 #. What is the area of the triangle's incircle?
- A circle has a center that falls on the line #y = 1/8x +4 # and passes through # ( 5 ,8 )# and #(5 ,6 )#. What is the equation of the circle?
- What are two formulas for finding the circumfrence of a circle?

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