question_answer

From a point within an equilateral triangle, perpendicular are drawn to its sides. The lengths of these perpendicular are 6 m, 7m and 8m. Find the area of the triangle

A)  160 sq. m                    

B)  \[147\sqrt{3}\]sq. m

C)  \[210\sqrt{3}\]sq. m.           

D)  \[27\sqrt{3}\]sq. nz

Correct Answer: B

Solution :

(b): Let a the side of an equilateral triangle ABC. Area of \[\Delta OBC+area\,of\Delta OAC+area\,of\Delta OAB\]\[=Area\,of\Delta ABC\] \[\frac{1}{2}\times a\times 6+\frac{1}{2}\times a\times 7+\frac{1}{2}\times a\times 8=\frac{\sqrt{3}}{4}{{a}^{2}}\] \[\frac{1}{2}a\times 21=\frac{\sqrt{3}}{4}{{a}^{2}}\] \[a=\frac{42}{\sqrt{3}}=\frac{42\times \sqrt{3}}{3}=14\sqrt{3}\] \[\therefore \] area of triangle \[=\frac{\sqrt{3}}{4}\times {{a}^{2}}\] \[=\frac{\sqrt{3}}{4}{{\left( 14\sqrt{3} \right)}^{2}}=\frac{\sqrt{3}}{4}\times 196\times 3=147\sqrt{3}\] sq.cm