Question

Consider a triangle ABC and let a,b and c denote the lengths of the sides opposite to vertices A,B,C respectively. Suppose a=6,b=10 and the area of the triangle is 15√3. If ∠ACB is obtuse and if r denotes the radius of the incircle of the triangle, then r² equals to

A. 3
B. 4
C. 5
D. 1

Answer 1

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.