Basic Mathematics & Logarithm
Inequalities involving means
Grade 11
Question:
<p>If \(A\) is the area and \(2s\) the sum of the sides of a triangle, then</p>
<p>(1) \(A \leq \dfrac{s^2}{4}\)</p>
<p>(2) \(A \leq \dfrac{s^2}{3\sqrt{3}}\)</p>
<p>(3) \(A < \dfrac{s^2}{\sqrt{3}}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: For any triangle, the area A relates to its semi-perimeter s through the inequality A ≤ s²/3√3, with equality only for equilateral triangles. This follows from A = √[s(s-a)(s-b)(s-c)] and the AM-GM inequality applied to the side differences.
<p><strong>Step 1:</strong> Let 2s be the sum of sides, so s is the semi-perimeter. By Heron's formula:</p><p>A = √[s(s-a)(s-b)(s-c)]</p><p><strong>Step 2:</strong> Apply AM-GM inequality to (s-a), (s-b), (s-c):</p><p>[(s-a)(s-b)(s-c)]^(1/3) ≤ [(s-a)+(s-b)+(s-c)]/3 = s/3</p><p><strong>Step 3:</strong> Therefore: A ≤ √[s · (s/3)³] = √[s⁴/27] = s²/(3√3)</p><p><strong>Step 4:</strong> Equality holds when s-a = s-b = s-c, meaning a = b = c (equilateral triangle).</p><p><strong>Step 5:</strong> For any triangle: A ≤ s²/(3√3) or equivalently <strong>3√3·A ≤ s²</strong></p><p>∴ Answer: AB (multiple valid inequalities expressing upper bound on area)</p>
Correct Answer: AB