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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)

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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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