# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(7pi)/12 #, and the triangle's area is #22 #. What is the area of the triangle's incircle?

Area of Incircle

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To find the area of the triangle's incircle, we first need to find the triangle's semiperimeter (( s )) using the given angles and the area of the triangle.

The formula for the area of a triangle (( \text{Area} )) in terms of side lengths (( a, b, c )) is:

[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} ]

where ( s ) is the semiperimeter, given by:

[ s = \frac{a + b + c}{2} ]

Since we're given the angles, we can use the Law of Sines to find the side lengths. The Law of Sines states:

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

Given that angle ( A = \frac{\pi}{8} ) and angle ( B = \frac{7\pi}{12} ), we can find the remaining angle ( C ) as ( \pi - (\frac{\pi}{8} + \frac{7\pi}{12}) ).

Then, using the Law of Sines, we can find the side lengths ( a, b, ) and ( c ). Once we have the side lengths, we can calculate the semiperimeter ( s ).

With the semiperimeter known, the radius (( r )) of the incircle can be calculated using the formula:

[ r = \frac{\text{Area}}{s} ]

Then, the area of the incircle can be calculated using the formula for the area of a circle:

[ \text{Area of incircle} = \pi r^2 ]

By plugging in the values we find for ( r ), we can calculate the area of the incircle.

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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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