Trigonometry & Inverse Trigonometry
Cosine Rule Application
Grade 11
Question:
<p>In triangle ABC, if AC = 8, BC = 7, and D lies between A and B such that AD = 2, BD = 4, then the length CD equals</p>
<p>(a) \(\sqrt{46}\)</p>
<p>(b) \(\sqrt{48}\)</p>
<p>(c) \(\sqrt{51}\)</p>
<p>(d) \(\sqrt{75}\)</p>
Step-by-Step Solution
Key Concept: Apply the cosine rule in triangle ACD after finding the angle A from triangle ABC
<p><strong>Step 1:</strong> In triangle ABC, use the cosine rule: \(AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos C\)</p><p><strong>Step 2:</strong> We have \(AB = AD + DB = 2 + 4 = 6\)</p><p><strong>Step 3:</strong> From triangle ACD, using the cosine rule: \(CD^2 = AC^2 + AD^2 - 2 \cdot AC \cdot AD \cdot \cos A\)</p><p><strong>Step 4:</strong> First find \(\cos A\) from the given sides: \(\cos A = \frac{BC^2 + AB^2 - AC^2}{2 \cdot BC \cdot AB} = \frac{49 + 36 - 64}{2 \cdot 7 \cdot 6} = \frac{21}{84} = \frac{1}{4}\)</p><p><strong>Step 5:</strong> Now, \(CD^2 = 64 + 4 - 2 \cdot 8 \cdot 2 \cdot \frac{1}{4} = 68 - 8 = 60\)</p><p>Wait, recalculating: \(CD^2 = 4 + 64 - 32\cos A = 68 - 32 \cdot \frac{17}{32} = 51\)</p><p>∴ \(CD = \sqrt{51}\), Answer is (c).</p>
Correct Answer: c