Question

If the angles of a triangle are in the ratio \[1:2:7\], then find the ratio of its greatest side to the least side.
A. \[1:2\]
B. \[2:1\]
C. \[\left( {\sqrt 5 + 1} \right):\left( {\sqrt 5 - 1} \right)\]
D. \[\left( {\sqrt 5 - 1} \right):\left( {\sqrt 5 + 1} \right)\]

Answer 1

Hint: In the given question, we need to find the ratio of greatest side to the least side of a triangle. For this, we need to assume the angles of a triangle such as \[1x,2x\] and \[7x\]. After that, we will use the property of a triangle such as the sum of all the angles of a triangle is \[{180^ \circ }\] to find the value of angles of a triangle. Based on the values of angles of a triangle, we will find the greatest and least side and will find the ratio of it using the following formula.

Formula used: The following formula used for solving the given question.
Suppose \[a,b\] and \[c\] are the sides of a triangle \[ABC\] and also \[A,B\] and \[C\] are the angles of a triangle \[ABC\] then \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Also, \[\angle A + \angle B + \angle C = {180^o}\]

Complete step by step solution:
We know that the angles of a triangle are in the ratio \[1:2:7\].
Let angles of a triangle be \[x,2x\] and \[7x\].
Let \[a,b\] and \[c\] are the sides of a triangle \[ABC\].
We know that the sum of all the angles of a triangle is \[{180^0}\].
Thus, \[x + 2x + 7x = {180^0}\]
By simplifying, we get
\[10x = {180^0}\]
Hence, \[x = {18^0}\]
So,\[\angle A = x = {18^o}\],\[\angle B = 2x = {36^o}\] and \[\angle C = 7x = {126^o}\]
Here, we can say that \[a\] is the least side while \[c\] is the greatest side.
So, \[\dfrac{a}{{\sin A}} = \dfrac{c}{{\sin C}}\]
Thus, we get \[\dfrac{a}{{\sin {{18}^o}}} = \dfrac{c}{{\sin {{126}^o}}}\]
So, \[\dfrac{c}{a} = \dfrac{{\sin {{126}^o}}}{{\sin {{18}^o}}}\]
By simplifying, we get
\[\dfrac{c}{a} = \dfrac{{\sin \left( {{{36}^o} + {{90}^o}} \right)}}{{\sin {{18}^o}}}\]
\[\dfrac{c}{a} = \dfrac{{\cos \left( {{{36}^o}} \right)}}{{\sin {{18}^o}}}\]
\[\dfrac{c}{a} = \dfrac{{\left( {\sqrt 5 + 1} \right)/4}}{{\left( {\sqrt 5 - 1} \right)/4}}\]
Hence, we get \[\dfrac{c}{a} = \dfrac{{\left( {\sqrt 5 + 1} \right)}}{{\left( {\sqrt 5 - 1} \right)}}\]

Therefore, the correct option is (C).

Note: Many students make mistakes in solving the calculation part and applying the right property of a triangle. This is the only way through which we can solve the example in the simplest way. Also, it is essential to write angles of trigonometric function properly to get the desired result.