question_answer

PQR is an equilateral triangle. O is the point of intersection of altitudes PL, QM RN, If \[\mathbf{OP}=\mathbf{9}\]cm, then what is the perimeter of the \[\Delta \mathbf{PQR}\]?

A)  \[6\sqrt{3}\]cm                         

B)  \[12\sqrt{3}\]cm       

C)  \[16\sqrt{3}\]cm          

D)  \[18\sqrt{3}\]cm.

Correct Answer: D

Solution :

(d):- Since, PQR is an equilateral. Then, PL is also the median of \[\Delta PQR\], Similarly RN and QM are also the median and 0 is the centroid. So, \[\frac{PO}{OL}=\frac{2}{1}\] \[OL=\frac{PO}{2}=\frac{9}{2}\]cm Now, Altitude of \[\Delta PQR=\frac{\sqrt{3x}}{2}\]  (Where, \[x=\] length of side of equilateral \[\Delta PQR\]) \[PO+OL=\frac{\sqrt{3a}}{2}\] \[6+3=\frac{\sqrt{3a}}{2}\] \[a=\frac{18}{\sqrt{3}}=6\sqrt{3}\] \[\therefore \] Perimeter of \[\Delta PQR=3a\] \[=3\times 6\sqrt{3}=18\sqrt{3}\]cm