<p>Consider an obtuse angled triangle with sides 8 cm, 15 cm and \(x\) cm (largest side being 15 cm). If \(x\) is an integer, then find the number of possible triangles.</p>
Step-by-Step Solution
Key Concept: Apply the cosine rule for obtuse angles where the cosine value lies between -1 and 0, combined with triangle inequality constraints.
<p><strong>Step 1:</strong> Since 15 is the largest side, we use the cosine rule: \(\cos\theta = \frac{x^2 + 8^2 - 15^2}{16x} = \frac{x^2 - 161}{16x}\)</p><p><strong>Step 2:</strong> For an obtuse angled triangle: \(-1 < \cos\theta < 0\)</p><p>Therefore: \(-16x < x^2 - 161 < 0\)</p><p><strong>Step 3:</strong> From \(x^2 - 161 < 0\): \(x^2 < 161\), so \(x < 13\) ... (i)</p><p><strong>Step 4:</strong> From \(x^2 + 16x - 161 > 0\): \((x + 23)(x - 7) > 0\), so \(x > 7\) ... (ii)</p><p><strong>Step 5:</strong> Combining conditions: \(7 < x < 13\)</p><p>Therefore: \(x \in \{8, 9, 10, 11, 12\}\)</p><p><strong>Answer:</strong> The number of possible values of \(x\) is <strong>5</strong>.</p>
Correct Answer: 5