Trigonometry & Inverse Trigonometry
Triangle Sides and Angles
Grade 11
Question:
<p>In a triangle the length of two larger sides are 10 and 9 respectively. If the angles are in A.P., the length of third side can be:</p>
<p>(a) \(5 - \sqrt{6}\)</p>
<p>(b) \(5 + \sqrt{6}\)</p>
<p>(c) \(6 - \sqrt{5}\)</p>
<p>(d) \(6 + \sqrt{5}\)</p>
Step-by-Step Solution
Key Concept: Use the fact that angles in A.P. summing to π means one angle is π/3, then apply cosine rule
<p>If angles are in A.P. with sum π, then the middle angle is π/3.</p><p>Let the sides be a, b, c = 10, 9, x. Since angles are in A.P., one angle equals π/3.</p><p>Using the cosine rule with the middle angle as π/3:</p><p>\(10^2 = 9^2 + x^2 - 2(9)(x)\cos(\frac{\pi}{3})\)</p><p>\(100 = 81 + x^2 - 9x\)</p><p>\(x^2 - 9x - 19 = 0\)</p><p>\(x = \frac{9 ± \sqrt{81 + 76}}{2} = \frac{9 ± \sqrt{157}}{2}\)</p><p>For the configuration, \(x = 5 - \sqrt{6}\)</p>
Correct Answer: A