<p>In \(\triangle ABC\), circumradius is 3 and inradius is 1.5 units. If the value of \(a\cot^2 A + b^2\cot^3 B + c^3\cot^4 C\) is \(m\sqrt{n}\) where m and n are prime numbers, then find the value of \(\left(\dfrac{m-1}{n}\right)\).</p>
Step-by-Step Solution
Key Concept: Use the relationship between circumradius R, inradius r, and triangle parameters: r = (s-a)tan(A/2) = 4R·sin(A/2)sin(B/2)sin(C/2). Convert cot expressions using cot θ = cos θ/sin θ and relate to R and r through projection formulas and area relationships.
<p><strong>Step 1:</strong> Use fundamental relations: R = 3, r = 1.5, and r/R = 1/2. From Euler's formula: OI² = R(R-2r), we get OI² = 3(3-3) = 0, meaning O = I. This occurs only for equilateral triangles.</p><p><strong>Step 2:</strong> For an equilateral triangle with R = 3: side length a = b = c = R√3 = 3√3. Also A = B = C = 60°, so cot 60° = 1/√3.</p><p><strong>Step 3:</strong> Calculate the expression:<br/>a·cot²A + b²·cot³B + c³·cot⁴C<br/>= 3√3·(1/√3)² + (3√3)²·(1/√3)³ + (3√3)³·(1/√3)⁴<br/>= 3√3·(1/3) + 27·(1/3√3) + 81√3·(1/9)<br/>= √3 + 9/√3 + 9√3<br/>= √3 + 3√3 + 9√3 = 13√3</p><p><strong>Step 4:</strong> We have m = 13 (prime) and n = 3 (prime), so m√n = 13√3.</p><p><strong>Step 5:</strong> Calculate (m-1)/n = (13-1)/3 = 12/3 = <strong>4</strong></p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4