Question

The area of an equilateral triangle is $576d{m}^2.$ The length of its median is

• A.
  $24\sqrt{3}dm$
• B.
  $24\sqrt[4]{43}dm$
• C.
  $12\sqrt[4]{43}dm$
• D.
  $12\sqrt[3]{34}dm$

Answer 1

The correct option is B

$24\sqrt[4]{43}dm$


Let ABC be an equilateral triangle and AD be its median.

$\therefore \text{Area of}\Delta ABC=\frac{\sqrt{3}}{4}(Side{)}^2\Rightarrow \frac{\sqrt{3}}{4}\times (Side{)}^2=576\Rightarrow Side=\sqrt{\frac{576\times 4}{\sqrt{3}}}\Rightarrow Side=\frac{48}{(3{)}^{\frac{1}{4}}}\therefore AB=AC=BC=\frac{48}{(3{)}^{\frac{1}{4}}}dm$

In an equilateral triangle, medians and altitudes are same. $Ar\left(\Delta ABC\right)=576\Rightarrow \frac{1}{2}\times BC\times AD=576\Rightarrow \frac{1}{2}\times \frac{48}{\sqrt[4]{43}}\times AD=576\Rightarrow AD=\frac{576}{24}\times \sqrt[4]{43}\Rightarrow AD=24\sqrt[4]{43}cm$

Hence, the correct answer is option (b).