# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(pi)/2 #, and the triangle's area is #12 #. What is the area of the triangle's incircle?

Area of incircle is

Let us consider a right angled triangle in general and an incircle within it as shown below.

Observe that centre of incircle

Here we have two angles

i.e.

Hence

and

and

Hence

and area of incircle is

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To find the area of the triangle's incircle, we can first calculate the lengths of the sides of the triangle using trigonometry. Given that vertex A has an angle of ( \frac{\pi}{12} ) and vertex B has an angle of ( \frac{\pi}{2} ), we can use the law of sines to relate the sides to these angles.

Let's denote the sides as ( a ), ( b ), and ( c ), where side ( a ) is opposite vertex A, side ( b ) is opposite vertex B, and side ( c ) is opposite vertex C.

Using the law of sines:

[ \frac{a}{\sin(\frac{\pi}{12})} = \frac{b}{\sin(\frac{\pi}{2})} = \frac{c}{\sin(\pi - \frac{\pi}{12} - \frac{\pi}{2})} ]

Simplifying, we have:

[ \frac{a}{\sin(\frac{\pi}{12})} = \frac{b}{1} = \frac{c}{\sin(\frac{11\pi}{12})} ]

Since ( \sin(\frac{\pi}{2}) = 1 ) and ( \pi - \frac{\pi}{12} - \frac{\pi}{2} = \frac{11\pi}{12} ).

Now, we can solve for ( c ) in terms of ( a ) using the relation between the sides and angles:

[ c = \frac{a \cdot \sin(\frac{11\pi}{12})}{\sin(\frac{\pi}{12})} ]

Next, we can find the semiperimeter ( s ) of the triangle using the formula:

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

Since we know the area of the triangle is 12, we can use Heron's formula to write:

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

Substitute the expressions for ( s ), ( a ), ( b ), and ( c ) into the equation above and solve for ( s ). Once you have ( s ), you can find the lengths of the sides ( a ), ( b ), and ( c ).

Finally, use the formula for the area of the incircle:

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

Substitute the known values to find the area of the triangle's 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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