# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/12 #, and the triangle's area is #12 #. What is the area of the triangle's incircle?

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The area of the triangle's incircle is (6\sqrt{3} - 3\pi).

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The area of a triangle can be expressed in terms of its semiperimeter (s) and the radii (r) of its incircle (r) and circumcircle (R) using the formula:

Area = s * r

where s = (a + b + c) / 2 and a, b, c are the lengths of the sides of the triangle.

The semiperimeter (s) can be found by adding the lengths of the sides of the triangle and dividing by 2:

s = (AB + BC + CA) / 2

To find the lengths of the sides AB, BC, and CA, you can use the law of sines:

AB = 2R * sin(A) BC = 2R * sin(B) CA = 2R * sin(C)

Given that the area of the triangle is 12 and the angles at vertices A and B are π/12 and (5π)/12 respectively, you can use these angles to calculate the lengths of the sides AB, BC, and CA. Once you have the lengths of the sides, you can find the semiperimeter (s) and then use it along with the area to find the radius of the incircle (r).

Finally, once you have the radius of the incircle (r), you can calculate the area of the incircle using the formula for the area of a circle:

Area of incircle = π * r^2

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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 chord that goes from #( 2 pi)/3 # to #(11 pi) / 12 # radians on the circle. If the area of the circle is #48 pi #, what is the length of the chord?
- A triangle has corners at #(1 ,3 )#, #(2 ,5 )#, and #(6 ,4 )#. What is the area of the triangle's circumscribed circle?
- A circle has a center that falls on the line #y = 8/9x +7 # and passes through # ( 4 ,1 )# and #(3 ,7 )#. What is the equation of the circle?
- A circle's center is at #(8 ,3 )# and it passes through #(2 ,5 )#. What is the length of an arc covering #( pi ) /6 # radians on the circle?
- A circle's center is at #(2 ,1 )# and it passes through #(6 ,1 )#. What is the length of an arc covering #(13pi ) /12 # radians on the circle?

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