
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
162.6k+ views
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.
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.
Recently Updated Pages
Fluid Pressure - Important Concepts and Tips for JEE

JEE Main 2023 (February 1st Shift 2) Physics Question Paper with Answer Key

Impulse Momentum Theorem Important Concepts and Tips for JEE

Graphical Methods of Vector Addition - Important Concepts for JEE

JEE Main 2022 (July 29th Shift 1) Chemistry Question Paper with Answer Key

JEE Main 2023 (February 1st Shift 1) Physics Question Paper with Answer Key

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

IIIT JEE Main Cutoff 2024

IIT Full Form

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

JEE Main Cut-Off for NIT Kurukshetra: All Important Details

JEE Main Cut-Off for VNIT Nagpur 2025: Check All Rounds Cutoff Ranks

Other Pages
NEET 2025: All Major Changes in Application Process, Pattern and More

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

NEET Total Marks 2025: Important Information and Key Updates

1 Billion in Rupees - Conversion, Solved Examples and FAQs

NEET 2025 Syllabus PDF by NTA (Released)

Important Days In June: What Do You Need To Know
