# A triangle has corners at #(3 , 3 )#, #(1 ,2 )#, and #(5 ,4 )#. What is the radius of the triangle's inscribed circle?

The three points lie on same straight line and hence no distinct triangle can be formed. In other words, one could say that radius of inscribed circle is

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To find the radius of the inscribed circle in the given triangle, we can use the formula:

[ r = \frac{A}{s} ]

Where:

- ( r ) is the radius of the inscribed circle.
- ( A ) is the area of the triangle.
- ( s ) is the semi-perimeter of the triangle.

First, we need to calculate the lengths of the sides of the triangle using the distance formula:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For the sides of the triangle, we have:

- Side 1: Between (3, 3) and (1, 2).
- Side 2: Between (1, 2) and (5, 4).
- Side 3: Between (5, 4) and (3, 3).

After calculating the lengths of the sides, we can find the semi-perimeter (( s )) by summing up the lengths of the three sides and dividing by 2.

Then, we'll use Heron's formula to find the area (( A )) of the triangle:

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

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

Once we have the area, we can find the radius (( r )) using the formula mentioned earlier.

Let's perform these calculations:

Side 1: (\sqrt{(1 - 3)^2 + (2 - 3)^2} = \sqrt{4 + 1} = \sqrt{5})

Side 2: (\sqrt{(5 - 1)^2 + (4 - 2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5})

Side 3: (\sqrt{(5 - 3)^2 + (4 - 3)^2} = \sqrt{4 + 1} = \sqrt{5})

Semi-perimeter: (s = \frac{\sqrt{5} + 2\sqrt{5} + \sqrt{5}}{2} = \frac{4\sqrt{5}}{2} = 2\sqrt{5})

Area: (A = \sqrt{2\sqrt{5}(2\sqrt{5}-\sqrt{5})(2\sqrt{5}-\sqrt{5})(2\sqrt{5}+\sqrt{5})} = \sqrt{2\sqrt{5} \cdot \sqrt{5} \cdot \sqrt{5} \cdot 3\sqrt{5}} = \sqrt{2 \cdot 5 \cdot 5 \cdot 3} = \sqrt{150} = 5\sqrt{6})

Now, using the formula (r = \frac{A}{s}):

(r = \frac{5\sqrt{6}}{2\sqrt{5}} = \frac{5\sqrt{6}}{2})

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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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