# Points #(2 ,4 )# and #(1 ,9 )# are #(3 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?

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To find the shortest arc length between two points on a circle, we need to compute the angular distance between these points and then use this angle to calculate the arc length.

The angular distance between two points on a circle is the angle subtended by the arc formed by connecting these points to the center of the circle. We can use the formula for the angle subtended by an arc in radians:

[ \text{angle} = \frac{\text{arc length}}{\text{radius}} ]

Given that the points (2, 4) and (1, 9) are ( \frac{3\pi}{4} ) radians apart, we can use the distance formula to find the distance between these points, which represents the arc length.

Using the distance formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

[ d = \sqrt{(1 - 2)^2 + (9 - 4)^2} ]

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

[ d = \sqrt{1 + 25} ]

[ d = \sqrt{26} ]

So, the arc length between the points (2, 4) and (1, 9) is ( \sqrt{26} ) 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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