question_answer

The measurement of three perpendiculars of an equilateral triangle is 10 cm, 12 cm and 14 cm, then find the length of side of equilateral triangle.

A)  \[24\sqrt{3}\,cm\]         

B)  \[24\,\,cm\]

C)  \[36\,\,cm\]      

D)  \[27\,\,cm\]

Correct Answer: A

Solution :

Let ABC be an equilateral triangle whose perpendiculars OD, OE and OF are 10 cm. 12 cm. and 14 cm respectively. Let the side of equilateral triangle AB, BC and AC = a cm. \[\therefore \]  Area of \[\Delta \,ABC=\frac{1}{2}\times AB\times OD+\frac{1}{2}\] \[\times AC\times OF+\frac{1}{2}\times BC\times OF\] \[\Rightarrow \,\,\,\frac{\sqrt{3}}{4}\times {{a}^{2}}=\frac{1}{2}\times a(OD+OE+OF)\] \[(\because \,\,AB=BC=AC=a\,cm)\] \[\Rightarrow \,\,\,\frac{\sqrt{3}}{4}\times {{a}^{2}}=\frac{a}{2}\,(10+12+14)\] \[\Rightarrow \,\,\frac{\sqrt{3}}{4}\times {{a}^{2}}=\frac{a}{2}\times 36\Rightarrow \frac{\sqrt{3}}{2}a=36\] \[\therefore \,\,\,\,a=\frac{36\times 2}{\sqrt{3}}=\underline{\mathbf{24}\sqrt{\mathbf{3}}\,\mathbf{cm}}\]