# Points #(2 ,1 )# and #(5 ,9 )# are #(3 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?

The length of the line segment, c, between the two points points is:

Because the line segment between the two points and two radii form a triangle, we can use the Law of Cosines to find the radius:

The arc length, s, is found using the following:

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To find the shortest arc length between the points, you can use the formula for arc length on a circle, which is ( s = r \cdot \theta ), where ( s ) is the arc length, ( r ) is the radius of the circle, and ( \theta ) is the angle subtended by the arc in radians.

First, you need to find the radius of the circle. You can do this by using the distance formula between the two points, which is ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ), where ( (x_1, y_1) ) and ( (x_2, y_2) ) are the coordinates of the two points.

Given that the points are (2, 1) and (5, 9), you can calculate the distance between them:

( d = \sqrt{(5 - 2)^2 + (9 - 1)^2} ) ( d = \sqrt{3^2 + 8^2} ) ( d = \sqrt{9 + 64} ) ( d = \sqrt{73} )

Now, since the points are ( \frac{3\pi}{4} ) radians apart on the circle, and we know that the circumference of a circle is ( 2\pi r ), you can find the angle ( \theta ) that corresponds to ( \frac{3\pi}{4} ) radians.

( \frac{\theta}{2\pi} = \frac{\frac{3\pi}{4}}{2\pi} ) ( \frac{\theta}{2\pi} = \frac{3}{8} )

( \theta = 2\pi \times \frac{3}{8} ) ( \theta = \frac{3\pi}{4} )

Now, you can calculate the arc length using the formula:

( s = r \cdot \theta ) ( s = \sqrt{73} \cdot \frac{3\pi}{4} ) ( s = \frac{3\pi\sqrt{73}}{4} )

So, the shortest arc length between the points is ( \frac{3\pi\sqrt{73}}{4} ) 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.

- A triangle has corners at #(9 , 2 )#, #(4 ,7 )#, and #(5 ,8 )#. What is the radius of the triangle's inscribed circle?
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- What is the equation of the circle with a center at #(-4 ,6 )# and a radius of #3 #?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/6 #, and the triangle's area is #9 #. What is the area of the triangle's incircle?
- A triangle has corners at #(5 , 2 )#, #(2 ,3 )#, and #(3 ,4 )#. What is the radius of the triangle's inscribed circle?

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