Prove that the greatest angle is 120

[utkarshakash]

Homework Statement

The sides of a triangle are x^2+3x+3, 2x+3,x^2+2x. Prove that the greatest angle of the triangle is 120°.

Homework Equations

The Attempt at a Solution

The greatest angle is the one opposite to the greatest side. But how to decide which one is greatest side? Also the sides are variable.

 

[ehild]

The side lengths must be positive numbers. Is x²+3x+3>x²+2x? Is x²+3x+3>2x+3?

ehild

 

[haruspex]

Assuming x > 0, it should be obvious which is the longest side. (But you can't say in general which is the shortest.)
Do you know the cosine rule relating lengths of sides of a triangle?

 

[Mark44]

Homework Statement

The sides of a triangle are x^2+3x+3, 2x+3,x^2+2x. Prove that the greatest angle of the triangle is 120°.

Homework Equations

The Attempt at a Solution

The greatest angle is the one opposite to the greatest side. But how to decide which one is greatest side? Also the sides are variable.

It depends on the value of x. If you graph y₁ = x^2 + 3x + 3, y₂ = x^2 + 2x, and y₃ = 2x + 3, two of the graphs are parabolas and one is a straight line. On the interval [0, 1], the largest values are for y₁ = x^2 + 3x + 3 and the smallest are for y₂ = x^2 + 2x. For the interval [-1, 0] the order is different.

 

[haruspex]

It depends on the value of x.

No, it turns out not to.

 

[Mark44]

It depends on the value of x.

No, it turns out not to.

By what I said, I meant that the values of the three expressions depend on x. As you said, one of the sides is the longest, but for other two sides, it depends on which interval you're looking at.

 

[I like Serena]

Turns out x has to be positive, otherwise one of the sides has a negative length.
And if x is positive, there is only one side that can be longest.

I found the results were even more impressive when I worked it through...