# 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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To find the area of the triangle's incircle, you can use the formula:

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

First, calculate the semiperimeter (( s )) of the triangle using the formula:

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

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

Since we have the angles of the triangle, we can use trigonometric ratios to find the side lengths. For example, in a right triangle with angle ( \frac{\pi}{2} ) and hypotenuse ( c ), ( c = \frac{AB}{\sin{\frac{\pi}{2}}} = AB ).

Using the same logic, we can find ( AB ) and ( BC ).

Then, calculate ( s ).

Next, use Heron's formula to find the area of the triangle using the side lengths:

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

Once you have the area of the triangle, you can rearrange the formula to solve for the inradius (( r )):

[ r = \frac{\text{Area}}{s} ]

Substitute the values of the area and semiperimeter into this equation to find the inradius, which represents the radius 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.

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

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