Final answer:
To find the radius of the incircle of the triangle, we use the formula for the area of a triangle and the formula for the semi-perimeter. By substituting the given values and solving the equation, we find that the radius squared (r^2) is equal to 1. Therefore, the correct answer is D. 1.
Step-by-step explanation:
To find the radius of the incircle of the triangle, we need to use the formula for the area of a triangle in terms of its side lengths. The formula is given by:
Area = (s * r) / 2
Where s is the semi-perimeter of the triangle and r is the inradius. We are given that the area of the triangle is 15√3, so we can substitute this value into the formula.
15√3 = (s * r) / 2
Now, let's use the formula for the semi-perimeter:
s = (a + b + c) / 2
Substituting the given values, we have:
s = (6 + 10 + c) / 2
Simplifying, we get:
s = (16 + c) / 2
Now, we can substitute the values of s and the area into the first equation and solve for r:
15√3 = ((16 + c) / 2) * r
Multiplying both sides by 2 and dividing by (16 + c), we get:
r = (30√3) / (16 + c)
Since ∠ACB is obtuse, c is greater than both a and b. So, c > 10. We can substitute this inequality into the equation for r:
r = (30√3) / (16 + c)
Since c > 10, 16 + c > 26. Therefore, the denominator is positive. This means that r is positive. Therefore, r^2 must be positive. So, the correct answer is D. 1.