# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #8 #. 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 in circle is

The area of the incircle is

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To find the area of the triangle's incircle, we can use the formula ( A = rs ), where ( A ) is the area of the triangle, ( r ) is the radius of the incircle, and ( s ) is the semi-perimeter of the triangle.

First, we need to find the lengths of the sides of the triangle using the given angles.

Let ( a ), ( b ), and ( c ) be the lengths of the sides opposite to angles ( A ), ( B ), and ( C ) respectively.

Using the Law of Sines, we have:

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

Given that ( A = \frac{\pi}{8} ) and ( B = \frac{\pi}{12} ), we can find ( C ) using the fact that the sum of angles in a triangle is ( \pi ).

[ C = \pi - A - B = \pi - \frac{\pi}{8} - \frac{\pi}{12} = \frac{5\pi}{24} ]

Now, we can find ( a ), ( b ), and ( c ) using the Law of Sines.

[ a = \frac{8}{\sin(\frac{\pi}{8})} ] [ b = \frac{8}{\sin(\frac{\pi}{12})} ] [ c = \frac{8}{\sin(\frac{5\pi}{24})} ]

Next, calculate the semi-perimeter ( s ) of the triangle:

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

Now, we can use Heron's formula to find the area ( A ) of the triangle:

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

Once we have found ( A ), we can calculate the radius ( r ) of the incircle using the formula ( r = \frac{A}{s} ).

Finally, the area of the incircle is given by ( \pi r^2 ). Calculate this value to find 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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- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/3 #, and the triangle's area is #24 #. What is the area of the triangle's incircle?

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