Question:
If the area of triangle \(ABC\) is \(4\sqrt{5}\) sq. units, length of the side \(CA\) is 6 units and \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\), then its smallest side is of length:
If the area of triangle \(ABC\) is \(4\sqrt{5}\) sq. units, length of the side \(CA\) is 6 units and \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\), then its smallest side is of length:
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Use half-angle and area formulas to find unknown sides in a triangle.
Updated On: Jun 4, 2025
- \(5\) units
- \(4\) units
- \(3\) units
- \(6\) units
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The Correct Option is C
Solution and Explanation
Step 1: Known values
Area \(= 4\sqrt{5}\), side \(CA = 6\), \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\). Step 2: Use half-angle formula
\[ \tan \frac{B}{2} = \sqrt{\frac{(s - a)(s - c)}{s(s - b)}} \] where \(a, b, c\) are sides opposite to vertices \(A, B, C\) and \(s\) is semi-perimeter. Step 3: Calculate semi-perimeter and use area
Using area formula and given data, find unknown sides and identify smallest. Step 4: Result
Smallest side length is \(3\) units.
Area \(= 4\sqrt{5}\), side \(CA = 6\), \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\). Step 2: Use half-angle formula
\[ \tan \frac{B}{2} = \sqrt{\frac{(s - a)(s - c)}{s(s - b)}} \] where \(a, b, c\) are sides opposite to vertices \(A, B, C\) and \(s\) is semi-perimeter. Step 3: Calculate semi-perimeter and use area
Using area formula and given data, find unknown sides and identify smallest. Step 4: Result
Smallest side length is \(3\) units.
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