Question : The ratio of the length of each equal side and the third side of an isosceles triangle is 3 : 5. If the area of the triangle is $30 \sqrt{11}$ cm2, then the length of the third side (in cm) is:
Option 1: $10\sqrt{6}$
Option 2: $5\sqrt{6}$
Option 3: $13\sqrt{6}$
Option 4: $11\sqrt{6}$
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Correct Answer: $10\sqrt{6}$
Solution : Let the equal sides be $3x$ cm and the third side be $5x$ cm. By heron's formula, Area of triangle = $\sqrt{s(s-a)(s-b)(s-c)}$ Where, $s=\frac{a+b+c}{2}$ Here, $s=\frac{3x+3x+5x}{2}$ ⇒ $s=\frac{11x}2$ So, $\sqrt{\frac{11x}2(\frac{11x}2-3x)(\frac{11x}2-3x)(\frac{11x}2-5x)} = 30\sqrt{11}$ ⇒ $\sqrt{\frac{11x}2(\frac{5x}2)(\frac{5x}2)(\frac{1x}2)} = 30\sqrt{11}$ ⇒ $\frac{5}{4}\sqrt{11}x^2 = 30\sqrt{11}$ ⇒ $x^2 = 24$ ⇒ $x=2\sqrt{6}$ So, third side = $5x=10\sqrt{6}$ cm Hence, the correct answer is $10\sqrt{6}$.
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