Question

If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the perimeter is

• A. $\sqrt { 3 } : ( 2 + \sqrt { 3 } )$
• B. 1 : 6
• C. $1 : ( 2 + \sqrt { 3 } )$
• D. $2 : 3$

Answer 1

Answer: (A)

$\sqrt { 3 } : ( 2 + \sqrt { 3 } )$

Answer 2

$4 x + x + x = 180$ ⇒ $6 x = 180$⇒ $x = 30 ^ { \circ }$

$\frac { \sin 120 ^ { \circ } } { a } = \frac { \sin 30 ^ { \circ } } { b } = \frac { \sin 30 ^ { \circ } } { c }$

$\therefore$ $a : ( a + b + c ) = \left( \sin 120 ^ { \circ } \right) : \left( \sin 120 ^ { \circ } + \sin 30 ^ { \circ } + \sin 30 ^ { \circ } \right)$

= $\frac { \sqrt { 3 } } { 2 } : \frac { \sqrt { 3 } + 2 } { 2 } = \sqrt { 3 } : \sqrt { 3 } + 2$.