# A triangle has sides with lengths of 2, 9, and 8. What is the radius of the triangles inscribed circle?

Radius of the inscribed circle

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The radius ( r ) of the inscribed circle of a triangle can be calculated using the formula:

[ r = \frac{{\text{{Area of the triangle}}}}{{\text{{Semiperimeter of the triangle}}}} ]

The semiperimeter ( s ) of the triangle is given by the formula:

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

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

Given that the sides of the triangle are ( a = 2 ), ( b = 9 ), and ( c = 8 ), we can find ( s ) as:

[ s = \frac{{2 + 9 + 8}}{2} = 9.5 ]

Next, we need to find the area ( A ) of the triangle using Heron's formula:

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

Substituting the values:

[ A = \sqrt{9.5(9.5 - 2)(9.5 - 9)(9.5 - 8)} = \sqrt{9.5 \times 7.5 \times 0.5 \times 1.5} = \sqrt{53.4375} \approx 7.31 ]

Now, we can find the radius ( r ) of the inscribed circle using the formula mentioned earlier:

[ r = \frac{A}{s} = \frac{7.31}{9.5} \approx 0.77 ]

Therefore, the radius of the triangle's inscribed circle is approximately ( 0.77 ) units.

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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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