# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/8 #, and the triangle's area is #15 #. 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 first need to find the lengths of the sides of the triangle. We can use the Law of Sines to find the lengths of the sides.

The Law of Sines states:

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

Given the angles A and B, we have:

( A = \frac{\pi}{12} ) and ( B = \frac{5\pi}{8} )

Since the sum of angles in a triangle is ( \pi ), we can find angle C:

( C = \pi - A - B )

Once we have the three angles, we can use the Law of Sines to find the ratio of the sides and then find the actual lengths of the sides using the given area.

After finding the lengths of the sides, we can calculate the semi-perimeter of the triangle, ( s ), and then use Heron's formula to find the area of the triangle.

Finally, with the area of the triangle known, we can use the formula for the area of a circle (( \pi r^2 )) to find the area of the incircle, where ( r ) is the radius of the incircle. The radius ( r ) can be found using the formula for the area of a triangle in terms of its inradius ( r ), which is ( A = rs ).

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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.

- 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 #9 #. What is the area of the triangle's incircle?
- A circle's center is at #(3 ,1 )# and it passes through #(6 ,6 )#. What is the length of an arc covering #(13pi ) /12 # radians on the circle?
- A circle has a center that falls on the line #y = 5/4x +5 # and passes through # ( 4 ,7 )# and #(2 ,5 )#. What is the equation of the circle?
- A triangle has corners at #(4 , 5 )#, #(3 ,2 )#, and #(5 ,3 )#. What is the radius of the triangle's inscribed circle?
- A circle has a chord that goes from #( pi)/2 # to #(13 pi) / 6 # radians on the circle. If the area of the circle is #4 pi #, what is the length of the chord?

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