# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/8 #, and the triangle's area is #5 #. 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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To find the area of the triangle's incircle, you can use the formula:

[ \text{Area} = \text{Semiperimeter} \times \text{Inradius} ]

where the semiperimeter is given by:

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

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

To find ( a ), ( b ), and ( c ), you can use the Law of Sines:

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

Given that the angles are ( \frac{\pi}{12} ), ( \frac{\pi}{8} ), and ( \frac{\pi}{3} ) (since the sum of angles in a triangle is ( \pi )), and the area is ( 5 ), you can solve for the sides of the triangle.

Once you have the sides of the triangle, you can calculate the semiperimeter. Then, you can use the formula for the area of the incircle to find the inradius. Finally, plug the semiperimeter and inradius into the formula for the area of the incircle to find the area.

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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 circle has a chord that goes from #( pi)/4 # to #(13 pi) / 8 # radians on the circle. If the area of the circle is #32 pi #, what is the length of the chord?
- A circle's center is at #(7 ,5 )# and it passes through #(2 ,7 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?
- A circle's center is at #(3 ,9 )# and it passes through #(5 ,8 )#. What is the length of an arc covering #(5pi ) /4 # radians on the circle?

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