# A triangle has sides with lengths of 2, 8, and 8. What is the radius of the triangles inscribed circle?

Refer to the figure below

Applying the Law of Sines

Because it is a triangle

Then

Sines of supplementary angles are equal or

So

As we can see from the figure

In case,

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The radius of the inscribed circle in a triangle can be found using the formula:

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

Where (A) is the area of the triangle and (s) is the semi-perimeter of the triangle.

Given the lengths of the sides of the triangle as 2, 8, and 8, we first need to find the semi-perimeter:

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

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

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

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

[ A = \sqrt{9(9 - 2)(9 - 8)(9 - 8)} ] [ A = \sqrt{9 \times 7 \times 1 \times 1} ] [ A = \sqrt{63} ]

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

[ r = \frac{\sqrt{63}}{9} ]

[ r = \frac{\sqrt{63}}{9} = \frac{3\sqrt{7}}{9} ]

[ r = \frac{\sqrt{7}}{3} ]

So, the radius of the triangle's inscribed circle is (\frac{\sqrt{7}}{3}).

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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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