# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/4 #, and the triangle's area is #15 #. What is the area of the triangle's incircle?

If 'r' is the radius of the incircle then

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The area of the triangle's incircle can be calculated using the formula:

[ \text{Area of incircle} = \frac{s}{2} \cdot r ]

Where ( s ) is the semi-perimeter of the triangle and ( r ) is the radius of the incircle. The semi-perimeter ( s ) can be calculated as:

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

Given that one angle of the triangle is ( \frac{\pi}{2} ) and another is ( \frac{\pi}{4} ), we can infer that the third angle is ( \frac{\pi}{4} ) as well, making it an isosceles right triangle. Let's denote the length of the sides opposite to the angles ( \frac{\pi}{4} ) as ( a ).

Then, using trigonometry, we find that ( a = b = c = \frac{15\sqrt{2}}{2} ). Now, we can calculate the semi-perimeter ( s ).

[ s = \frac{a + b + c}{2} = \frac{3 \cdot 15\sqrt{2}}{2} ]

Now, we can find the radius of the incircle using the formula:

[ r = \frac{\text{Area of triangle}}{s} = \frac{15}{\frac{3 \cdot 15\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} ]

Finally, we calculate the area of the incircle:

[ \text{Area of incircle} = \frac{s}{2} \cdot r = \frac{\frac{3 \cdot 15\sqrt{2}}{2}}{2} \cdot \sqrt{2} = \frac{45}{4} ]

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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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