# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/6 #, and the triangle's area is #98 #. What is the area of the triangle's incircle?

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To find the area of the triangle's incircle, follow these steps:

- Use the formula for the area of a triangle: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ).
- Identify the base and height of the triangle.
- Use the formula for the radius of the incircle of a triangle: ( r = \frac{\text{Area of the triangle}}{\text{Semi-perimeter of the triangle}} ).
- Calculate the semi-perimeter of the triangle: ( s = \frac{a + b + c}{2} ), where ( a ), ( b ), and ( c ) are the lengths of the triangle's sides.
- Calculate the radius ( r ) of the incircle using the formula from step 3.
- Use the formula for the area of the incircle of a triangle: ( \text{Area of incircle} = \pi \times r^2 ).

Given that the triangle has a right angle at vertex A, we can use trigonometric relationships to find the lengths of the triangle's sides.

Let's denote the sides as follows:

- ( AB ) is the side opposite the angle ( \frac{\pi}{6} ).
- ( AC ) is the side opposite the right angle ( \frac{\pi}{2} ).
- ( BC ) is the hypotenuse.

From the given information, we know that the area of the triangle is 98. We also know that the base of the triangle is ( AB ) and the height is ( AC ).

Using trigonometric relationships in a (30^\circ)-(60^\circ)-(90^\circ) triangle, we find that ( AB = AC\sqrt{3} ). Thus, the base and height are equal.

Using the area formula, we have: [ 98 = \frac{1}{2} \times AB \times AC ]

Since ( AB = AC\sqrt{3} ), we can rewrite the equation as: [ 98 = \frac{1}{2} \times (AC\sqrt{3}) \times AC ]

Solving for ( AC ), we get: [ AC^2 = \frac{98 \times 2}{\sqrt{3}} ]

Then, calculate the semi-perimeter ( s ) using the lengths of the sides ( AB ) and ( AC ).

Next, compute the radius ( r ) of the incircle using the formula mentioned above.

Finally, use the formula for the area of the incircle to find the answer.

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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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