# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/4 #, and the triangle's area is #3 #. What is the area of the triangle's incircle?

Using (1) and (6) we write:

Now use the Triangle Area Formula with r:

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To find the area of the triangle's incircle, you 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.

The semi-perimeter ( s ) can be calculated as:

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

where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.

Given that the angles at vertices A and B are ( \frac{\pi}{12} ) and ( \frac{\pi}{4} ) respectively, we can find the lengths of the sides using trigonometric ratios.

Let's denote the side opposite vertex A as ( a ), opposite vertex B as ( b ), and opposite vertex C as ( c ). We'll use the fact that ( \tan(\frac{\pi}{12}) = \frac{a}{3} ) and ( \tan(\frac{\pi}{4}) = \frac{b}{3} ).

We find: [ a = 3 \tan\left(\frac{\pi}{12}\right) ] [ b = 3 \tan\left(\frac{\pi}{4}\right) ]

Now, calculate ( c ) using the triangle inequality:

[ c < a + b ]

Given that the triangle's area is 3, we know ( A = 3 ).

Substituting the values of ( a ) and ( b ), we have:

[ 3 = rs ] [ r = \frac{3}{s} ]

We need to find ( s ), which is the semi-perimeter of the triangle.

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

Once you have ( s ), you can calculate ( r ) using the formula ( r = \frac{A}{s} ), where ( A ) is the area of the triangle. Then, once you have ( r ), you can find the area of the incircle using the formula for the area of a circle: ( A_{\text{incircle}} = \pi 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.

- What is the equation of the circle with a center at #(2 ,2 )# and a radius of #3 #?
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- A circle has a center that falls on the line #y = 7/2x +3 # and passes through #(1 ,2 )# and #(6 ,1 )#. What is the equation of the circle?
- A triangle has corners at #(9 ,4 )#, #(3 ,2 )#, and #(5 ,2 )#. What is the area of the triangle's circumscribed circle?
- A circle has a chord that goes from #pi/4 # to #(3 pi) / 8 # radians on the circle. If the area of the circle is #81 pi #, what is the length of the chord?

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