question_answer

PQR is an equilateral triangle. O is the point of intersection of altitudes PL, QM and RN. If \[OP=8\text{ }cm,\]then what is the perimeter of the triangle PQR?

A)  \[8\sqrt{3}\,\,cm\]                        

B)  \[12\sqrt{3}\,\,cm\]

C)  \[16\sqrt{3}\,\,cm\]                      

D)  \[24\sqrt{3}\,\,cm\]

Correct Answer: D

Solution :

 Since, PQR is an equilateral triangle Then, PL is also the median of . Similarly, RN and QM are also the median and O is the centroid. So, \[\frac{PO}{OL}=\frac{2}{1}\,\Rightarrow PL=\frac{PO}{2}=\frac{8}{2}=4\,cm\] Now, altitude of \[\Delta \,PQR=\frac{\sqrt{3}\,a}{2}\] (where, a = length of the side of equilateral triangle PQR) \[PO+OL=\frac{\sqrt{3}a}{2}\Rightarrow 8+4=\frac{\sqrt{3}a}{2}\] \[a=\frac{12\times 2}{\sqrt{3}}=\frac{24}{\sqrt{3}}\,\,cm\]                 \[\therefore \]  Perimeter of \[\Delta \text{ }PQR=3a\] \[=\frac{3\times 24}{\sqrt{3}}=\,\]