A triangle has sides with lengths of 2, 3, and 8. What is the radius of the triangles inscribed circle?

Given :sides of the triangle #a = 2, b = 3, c = 8.

To find the radius of the inscribed circle.

For a triangle to exist, sum of any two sides must be greater than the third side.

But in the given sum,

Hence such a triangle cannot exist.

To find the radius ( r ) of the inscribed circle in a triangle with side lengths ( a ), ( b ), and ( c ), you can use the formula:

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

Where ( \Delta ) is the area of the triangle. You can calculate the area ( \Delta ) using Heron's formula:

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

Where ( s ) is the semi-perimeter of the triangle, given by:

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

For the given triangle with side lengths 2, 3, and 8, the semi-perimeter is ( s = \frac{2 + 3 + 8}{2} = \frac{13}{2} ). Using Heron's formula, we find ( \Delta ). Then, we can find the radius ( r ) using the first formula.

[ \Delta = \sqrt{\frac{13}{2}\left(\frac{13}{2} - 2\right)\left(\frac{13}{2} - 3\right)\left(\frac{13}{2} - 8\right)} ]

[ \Delta = \sqrt{\frac{13}{2} \times \frac{9}{2} \times \frac{7}{2} \times \frac{5}{2}} ]

[ \Delta = \sqrt{\frac{13 \times 9 \times 7 \times 5}{2 \times 2 \times 2 \times 2}} ]

[ \Delta = \sqrt{\frac{13 \times 9 \times 7 \times 5}{8}} ]

[ \Delta = \sqrt{\frac{4095}{8}} ]

[ \Delta \approx 18.013 ]

Now, we can find the radius ( r ):

[ r = \frac{2 \times 18.013}{2 + 3 + 8} ]

[ r = \frac{36.026}{13} ]

[ r \approx 2.774 ]

So, the radius of the inscribed circle in the triangle with side lengths 2, 3, and 8 is approximately ( 2.774 ) units.