Question
Angles of a triangle are in the ratio $$4 : 1 : 1.$$ The ratio between its greatest side and perimeter is
A.
$$\frac{3}{{2 + \sqrt 3 }}$$
B.
$$\frac{1}{{2 + \sqrt 3 }}$$
C.
$$\frac{\sqrt 3}{{\sqrt 3 + 2 }}$$
D.
$$\frac{2}{{2 + \sqrt 3 }}$$
Answer :
$$\frac{\sqrt 3}{{\sqrt 3 + 2 }}$$
Solution :
Consider a triangle $$ABC.$$
Given, angles of a triangle are in the ratio $$4 : 1 : 1.$$
Angles are $$4x, x$$ and $$x.$$
$${\text{i}}{\text{.e}}{\text{., }}\angle A = 4x,\angle B = x,\angle C = x$$
Now, by angle sum property of $$\Delta ,$$ we have
$$\eqalign{
& \angle A + \angle B + \angle C = {180^ \circ } \cr
& \Rightarrow 4x + x + x = {180^ \circ } \cr
& \Rightarrow x = \frac{{{{180}^ \circ }}}{6} = {30^ \circ } \cr
& \therefore \angle A = {120^ \circ },\angle B = {30^ \circ },\angle C = {30^ \circ } \cr} $$
We know, ratio of sides of $$\Delta \,ABC$$ is given by
$$\eqalign{
& \sin A:\sin B:\sin C = \sin {120^ \circ }:\sin {30^ \circ }:\sin {30^ \circ } \cr
& = \frac{{\sqrt 3 }}{2}:\frac{1}{2}:\frac{1}{2} = \sqrt 3 :1:1 \cr} $$
Required ration $$ = \frac{{\sqrt 3 }}{{1 + 1 + \sqrt 3 }} = \frac{{\sqrt 3 }}{{2 + \sqrt 3 }}.$$