# A triangle has sides with lengths of 3, 9, and 8. What is the radius of the triangles inscribed circle?

Where s is the sum of all sides / 2 (Also known as the semiperimeter) a,b,c are the side lengths

Now, we use the formula that says that

Area = semiperimeter * inradius

So we have

We can simplify the radical and cancel out to get

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The radius of the triangle's inscribed circle can be calculated using the formula:

[ r = \frac{2 \times \text{Area}}{ \text{Perimeter}} ]

where ( r ) is the radius of the inscribed circle, Area is the area of the triangle, and Perimeter is the sum of the lengths of the sides of the triangle.

First, we need to find the semi-perimeter of the triangle, which is half of the perimeter:

[ s = \frac{3 + 9 + 8}{2} = 10 ]

Now, we can use Heron's formula to find the area of the triangle:

[ \text{Area} = \sqrt{s(s - 3)(s - 9)(s - 8)} ]

[ \text{Area} = \sqrt{10(10 - 3)(10 - 9)(10 - 8)} ]

[ \text{Area} = \sqrt{10(7)(1)(2)} ]

[ \text{Area} = \sqrt{140} ]

[ \text{Area} ≈ 11.83 ]

Now, we can calculate the radius of the inscribed circle:

[ r = \frac{2 \times 11.83}{3 + 9 + 8} ]

[ r = \frac{2 \times 11.83}{20} ]

[ r ≈ 1.18 ]

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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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