Triangles And Properties Of Triangle Question 29
Question: Angles of a triangle are in the ratio $ 4:1:1. $ The ratio between its greatest side and perimeter is
Options:
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}} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Consider a triangle ABC. Given, angles of a triangle are in the ratio $ 4:1:1. $ angles are 4x, x and x i.e., $ \angle A=4x,\angle B=x,\angle C=x $ Now, by angle sum property of $ \Delta $ , we have $ \angle A+\angle B+\angle C=180{}^\circ $
$ \Rightarrow 4x+x+x=180{}^\circ \Rightarrow x=\frac{180{}^\circ }{6}=30{}^\circ $
$ \therefore \angle A=120{}^\circ ,\angle B=30{}^\circ ,\angle C=30{}^\circ $ We know, ratio of sides of $ \Delta ABC $ is given by $ \sin A:\sin B:sinC=sin120{}^\circ :sin30{}^\circ :sin30{}^\circ $ $ =\frac{\sqrt{3}}{2}:\frac{1}{2}:\frac{1}{2}=\sqrt{3}:1:1 $ Required ratio $ =\frac{\sqrt{3}}{1+1+\sqrt{3}}=\frac{\sqrt{3}}{2+\sqrt{3}}. $
BETA
