Trigonometry & Inverse Trigonometry
Properties of Triangle
Grade 11
Question:
<p>If A is the area and 2S the sum of sides of a triangle, then</p>
<p>(a) \(A < \dfrac{S^2}{4}\)</p>
<p>(b) \(A \leq \dfrac{S^2}{3\sqrt{3}}\)</p>
<p>(c) \(A > \dfrac{S^2}{\sqrt{3}}\)</p>
<p>(d) \(A \leq \dfrac{S^2}{\sqrt{3}}\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental relation between area, semi-perimeter, and inradius: A = rS, where r is the inradius and S is the semi-perimeter. This connects area to the sum of sides through the inscribed circle.
<p><strong>Step 1:</strong> Identify that 2S is the sum of all three sides, so S is the semi-perimeter (half the sum of sides).</p><p><strong>Step 2:</strong> Recall the fundamental formula: A = rS, where r is the inradius and S is the semi-perimeter. This can be rearranged to: r = A/S.</p><p><strong>Step 3:</strong> The relationship between area A and sum of sides 2S is: A ≤ (S²√3)/3 (with equality when triangle is equilateral), or more directly: <strong>A = rS where r ≤ S/(3√3)</strong>.</p><p><strong>Step 4:</strong> The most common form tested is: <strong>A ≤ S²/(3√3)</strong> or equivalently <strong>3√3·A ≤ S²</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B