# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(3pi)/8 #, and the triangle's area is #6 #. What is the area of the triangle's incircle?

The area of the incircle is

The area of the triangle is

The angle

The angle

The angle

The sine rule is

So,

Let the height of the triangle be

The area of the triangle is

But,

So,

Therefore,

The radius of the incircle is

The area of the incircle is

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We can use the formula for the area of a triangle in terms of its sides and semiperimeter, which is given by:

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

where ( s ) is the semiperimeter of the triangle, and ( a ), ( b ), and ( c ) are the lengths of its sides. Since we know the area of the triangle is 6, we can write:

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

Now, let's denote the sides of the triangle as ( a ), ( b ), and ( c ). The angles ( \pi/12 ) and ( 3\pi/8 ) correspond to the angles at vertices A and B, respectively. Using the law of sines, we can relate the sides to these angles:

[ \frac{a}{\sin(\frac{\pi}{12})} = \frac{b}{\sin(\frac{3\pi}{8})} = \frac{c}{\sin(\pi - \frac{\pi}{12} - \frac{3\pi}{8})} ]

Now, solve for ( c ) in terms of ( a ) and ( b ) using the relationship between the sides and the angles:

[ c = \frac{b \cdot \sin(\pi - \frac{\pi}{12} - \frac{3\pi}{8})}{\sin(\frac{3\pi}{8})} ]

Next, substitute the expression for ( c ) into the formula for the area of the triangle:

[ 6 = \sqrt{s(s-a)(s-b)\left(s - \frac{b \cdot \sin(\pi - \frac{\pi}{12} - \frac{3\pi}{8})}{\sin(\frac{3\pi}{8})}\right)} ]

Solve this equation for ( s ), the semiperimeter of the triangle. Once you have ( s ), you can find the lengths of the sides ( a ), ( b ), and ( c ). Then, use Heron's formula to find the area of the triangle's incircle, which is given by:

[ \text{Area of incircle} = \frac{\text{Area of triangle}}{s} ]

Finally, substitute the values you found for the semiperimeter ( s ) and the area of the triangle into this formula to 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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