Question : The lengths of the three medians of a triangle are $9\;\mathrm{cm}$, $12\;\mathrm{cm}$, and $15\;\mathrm{cm}$. The area (in $\mathrm{cm^2}$) of the triangle is:
Option 1: $24$
Option 2: $72$
Option 3: $48$
Option 4: $144$
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Correct Answer: $72$
Solution :
The area of a triangle in terms of its medians,
Area $=\frac{4}{3} \sqrt{s(s - m_1)(s - m_2)(s - m_3)}$
where $m_1$, $m_2$ and $m_3$ are the lengths of the medians, and $s$ is the semi-perimeter of the medians, which is
$s = \frac{m_1 + m_2 + m_3}{2}$
The lengths of the medians are $9\;\mathrm{cm}$, $12\;\mathrm{cm}$ and $15\;\mathrm{cm}$.
$s = \frac{9 + 12 + 15}{2} =18\;\mathrm{cm}$
Area $=\frac{4}{3} \sqrt{18(18 - 9)(18 - 12)(18 - 15)}=\frac{4}{3} ×9×6= 72 \text{ cm}^2$
Hence, the correct answer is $72$.
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