IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

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The sides of a triangle are in GP..... will any angle of the triangle be obtuse??

skanda.b

no there is no possibility to get any obtuse angle when sides of the triangle are in gp

ramyadiamond

I dont think thats possible, try it out like this.

In a triangle ABC, lets suppose B is obtuse, say 2pi/3. Let sides of the triangle be a=m, b=mr, and c=mr². Now, apply cosine rule,

cos(2pi/3) =a²+c²-b²/2ac

-1/2 = m² + m²r⁴ - m²r²/2m²r²

-1/2*2r² = r⁴ -r²+1

which gives,

r⁴= -1

which is not possible, so i dont think there might be any triangle of that sort.

iitkgp_bipin

The side opposite to the obtuse angle(if any) would be the largest side.

Let the sides in increasing order be a,ar,ar² with r > 1 and let C be the largest angle of the triangle.

Then cosC = (a² + b² - c²)/(2ab) = {a² + (ar)² - (ar²)²}/{2(a)(ar)}

cosC = (1 + r² - r⁴)/r

Now analyze the numerator (1 + r² - r⁴) = 5/4 - (r² - 1/2)²

Hence for r > (5^1/2 + 1)/2 , the numerator is negative and hence cosC < 0

This means that the angle C can be obtuse.

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