Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>In a triangle ABC, if <span class='latex'>(a + b + c)(b + c - a) = Xbc</span>, where <span class='latex'>X \in I</span>, then the greatest value of X is:</p>
Step-by-Step Solution
Key Concept: Use properties of semi-perimeter and AM-GM inequality to find the maximum value.
<p><strong>Analysis:</strong> Given: <span class='latex'>(a + b + c)(b + c - a) = Xbc</span></p><p>Let <span class='latex'>s = \frac{a+b+c}{2}</span> (semi-perimeter). Then <span class='latex'>b + c - a = 2(s - a)</span></p><p>So: <span class='latex'>2s \cdot 2(s-a) = Xbc</span></p><p><span class='latex'>4s(s-a) = Xbc</span></p><p>Using the formula for area: <span class='latex'>\Delta = \sqrt{s(s-a)(s-b)(s-c)}</span> and other identities:</p><p><span class='latex'>X = \frac{4s(s-a)}{bc}</span></p><p>By AM-GM inequality and optimization, the maximum value of X is 3.</p><p>∴ Answer is (p) X = 3</p>
Correct Answer: p