A triangle has sides with lengths of 2, 6, and 8. What is the radius of the triangles inscribed circle?

Unfortunately, you cannot form a triangle with sides of 2, 6 & 8.

According to the Hinge Theorem, the two sides of a triangle must be greater than the third side. Since 2 + 6 = 8 this would not form a triangle.

Therefore there can be no circle inscribed and therefore no radius.

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

[ r = \frac{{\text{{Area of the triangle}}}}{{\text{{Semi-perimeter of the triangle}}}} ]

where the semi-perimeter of the triangle ( s ) is calculated as:

[ s = \frac{{\text{{sum of the lengths of the sides}}}}{2} ]

The area of the triangle can be found using Heron's formula:

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

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

Given that the lengths of the sides are ( a = 2, b = 6, ) and ( c = 8 ), we can find the semi-perimeter ( s ):

[ s = \frac{{2 + 6 + 8}}{2} = 8 ]

Then, using Heron's formula, we find the area of the triangle:

[ \text{{Area}} = \sqrt{8(8 - 2)(8 - 6)(8 - 8)} = \sqrt{8 \cdot 6 \cdot 2 \cdot 0} = 0 ]

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

[ r = \frac{{\text{{Area}}}}{{s}} = \frac{0}{8} = 0 ]

Therefore, the radius of the inscribed circle in the triangle is 0.