If 7 and 8 are the lengths of two sides of a triangle and ' $a$ ' is the length of its smallest side. The angles of the triangle are in AP and ' $a$ ' has two values $a_1$ and $a_2$ satisfying this condition. If $a_1 \lt a_2$ then $2 a_1+3 a_2=$

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Aicharya · Accepted Answer

Let angles be $A, 180^{\circ}-3 A, 2 A$
Then $2\left(180^{\circ}-3 A\right)=A+2 A \Rightarrow A=40$ $\therefore$ Angles are $40^{\circ}, 60^{\circ}, 80^{\circ}$ Now, $b^2=c^2+a^2-2 a c \cos 60^{\circ} \Rightarrow 49=64+a^2-8 a$ $\begin{aligned} & \Rightarrow a^2-8 a+15=0 \Rightarrow a=3,5 \\ & \Rightarrow a_1=3, a_2=5 \Rightarrow 2 a_1+3 a_2=21 \end{aligned}$

If 7 and 8 are the lengths of two sides of a triangle and ' $a$ ' is the length of its smallest side. The…

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