If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is : 

  • Option 1)

  • Option 2)

  • Option 3)

  • Option 4)

 

Answers (1)

Let a<b<c be side of   \bigtriangleup le 

\theta is smallest angle

Three angles are, \theta ,\pi -3\theta ,2\theta .

Given, 2b=a+c

Use sine rule.

2\sin \left ( B \right )=\sin \left ( A \right )+\sin \left ( C \right )

2\sin \left (3\theta \right )=\sin \left ( \theta \right )+\sin \left (2\theta \right )

2\left ( 3\sin \theta -4\sin ^{3}\theta \right )=\sin \theta \left ( 1+2\cos \theta \right )

6-8\left ( 1-\cos ^{2}\theta \right )=1+2\cos \theta

\cos \theta =\frac{3}{4},-\frac{1}{2}\left ( -\frac{1}{2}\: \: \: is\: \: \: rejected \right )

\therefore a:b:c=\sin A:\sin B:\sin C

                    =\sin \theta :\sin 3\theta :\sin 2\theta

                    =1:3-4\sin ^{2}\theta :2\cos \theta =1:4\cos ^{2}\theta -1:2\cos\theta

                   =4:5:6


Option 1)

Option 2)

Option 3)

Option 4)

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