Properties and Solutions of Triangles
AP Condition on Sides
Grade 11
Question:
<p>The sides of a triangle are in AP. If the angles A and C are the greatest and smallest angle respectively, then \(4(1 - \cos A)(1 - \cos C)\) is equal to</p>
<p>(a) \(\cos A - \cos C\)</p>
<p>(b) \(\cos A \cos C\)</p>
<p>(c) \(\cos A + \cos C\)</p>
<p>(d) \(\cos C - \cos A\)</p>
Step-by-Step Solution
Key Concept: Since the sides are in AP, express them as (a-d), a, (a+d) and use the relationship between sides and angles. Since A is greatest and C is smallest, the sides opposite to them follow: c < b < a. Apply cosine rule and algebraic manipulation to simplify the expression.
<p><strong>Step 1:</strong> Set up the sides in AP. Let the sides be b-d, b, b+d where d > 0. Since A is the greatest angle and C is the smallest angle, we have a > b > c. Therefore: a = b+d, b = b, c = b-d.</p><p><strong>Step 2:</strong> Apply the cosine rule: cos A = (b² + c² - a²)/(2bc) and cos C = (a² + b² - c²)/(2ab).</p><p><strong>Step 3:</strong> Substitute the sides. cos A = [b² + (b-d)² - (b+d)²]/[2b(b-d)] = [b² + b² - 2bd + d² - b² - 2bd - d²]/[2b(b-d)] = [b² - 4bd]/[2b(b-d)] = [b - 4d]/[2(b-d)].</p><p><strong>Step 4:</strong> Similarly, cos C = [(b+d)² + b² - (b-d)²]/[2b(b+d)] = [b² + 2bd + d² + b² - b² + 2bd - d²]/[2b(b+d)] = [b² + 4bd]/[2b(b+d)] = [b + 4d]/[2(b+d)].</p><p><strong>Step 5:</strong> Calculate (1 - cos A): 1 - cos A = 1 - [b - 4d]/[2(b-d)] = [2(b-d) - b + 4d]/[2(b-d)] = [b + 2d]/[2(b-d)].</p><p><strong>Step 6:</strong> Calculate (1 - cos C): 1 - cos C = 1 - [b + 4d]/[2(b+d)] = [2(b+d) - b - 4d]/[2(b+d)] = [b - 2d]/[2(b+d)].</p><p><strong>Step 7:</strong> Compute 4(1 - cos A)(1 - cos C) = 4 · [b + 2d]/[2(b-d)] · [b - 2d]/[2(b+d)] = 4 · [(b + 2d)(b - 2d)]/[4(b-d)(b+d)] = [(b² - 4d²)]/[(b² - d²)].</p><p><strong>Step 8:</strong> Calculate cos A + cos C = [b - 4d]/[2(b-d)] + [b + 4d]/[2(b+d)] = [1]/[2] · {[b - 4d]/[b-d] + [b + 4d]/[b+d]}.</p><p><strong>Step 9:</strong> Simplify: = [1]/[2] · {[(b - 4d)(b+d) + (b + 4d)(b-d)]/[(b-d)(b+d)]} = [1]/[2] · {[b² - 3bd - 4d² + b² + 3bd - 4d²]/[(b² - d²)]} = [1]/[2] · {[2b² - 8d²]/[(b² - d²)]} = [b² - 4d²]/[b² - d²].</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C