If the sides of a triangle are in AP, and the greatest angle of the triangle is double the smallest angle, the ratio of the sides of the triangle is

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Infinity Learn · Accepted Answer

Let the sides of the triangle be $$a-d$$, a and $$a+d$$, with a > d>$$0$$. Clearly, a $$-d$$ is the smallest side and a $$+$$ d is the largest side. So, A is the smallest angle and C is the largest angle. It is given that C $$=$$ $$2A$$. Thus, the angles of the triangle are A, $$2A$$ and  π$$-3A$$. Applying the law of sines, we obtain  a−dsin⁡$$A=asin$$⁡$$($$π−$$3A)=a+dsin$$⁡$$2A$$⇒a−dsin⁡$$A=asin$$⁡$$3A=a+dsin$$⁡$$2A$$⇒a−dsin⁡$$A=a3sin$$⁡A−$$4sin3$$⁡$$A=a+d2sin$$⁡Acos⁡A⇒a−$$d1=a3$$−$$4sin2$$⁡$$A=a+d2cos$$⁡A⇒$$3$$−$$4sin2$$⁡$$A=aa$$−$$dand2cos$$⁡$$A=a+da$$−d⇒$$4cos2$$⁡A−$$1=aa$$−$$dand2cos$$⁡$$A=a+da$$−d⇒ $$a+da$$−$$d2$$−$$1=aa$$−d⇒$$a=5dThus$$, the sides of the triangle are $$a-d$$, a, a $$+$$ d    i.e. $$4d$$, $$5d$$, $$6d$$.Hence, the ratio of the sides of the triangle is $$4d$$ :$$5d$$: $$6d$$     i.e. $$4$$:$$5$$:$$6$$.

If the sides of a triangle are in AP, and the greatest angle of the triangle is double the smallest angle, the ratio of the sides of the triangle is

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