Determining the Largest Angle in a Triangle with Given Side Lengths

Given the side lengths of a triangle represented by expressions involving variable x, it is possible to determine the largest angle using the side opposite the longest side. This guide details the process of identifying the longest side and, consequently, the largest angle in a triangle with side lengths defined by the expressions:

Side Lengths

The side lengths are given by:

a x2 - x 1 b 2x - 1 c x2 - 1

Determining the Largest Side

To identify the longest side, we will compare the expressions for a, b, and c to find the maximum value.

Compare a and b

To find the difference between a and b, we compute:

a - b  x^2 - x   1 - (2x - 1)  x^2 - 3x   2  (x - 1)(x - 2)

The expression a - b is positive when x 1 or x 2, and negative when 1 x 2. Therefore, a is greater than b if x 1 or x 2.

Compare a and c

The difference between a and c is:

a - c  x^2 - x   1 - (x^2 - 1)  x   2

This is always positive for all x -2. Therefore, a is always greater than c.

Compare b and c

The difference between b and c is:

b - c  2x - 1 - (x^2 - 1)  -x^2   2x  -(x^2 - 2x)  -(x - 1)^2   1

This is a downward-opening parabola with its vertex at x 1 and a maximum value of 1. Therefore, b is greater than c for x 1 - sqrt{3} or x 1 sqrt{3}, and c is greater than b for 1 - sqrt{3} x 1 sqrt{3}.

Conclusion

Based on the comparisons, the side length a is the longest when:

x 1 or x 2 x 1

The side length c is the longest when:

1 x 2

Therefore, the largest angle of the triangle is opposite side a if x 1 or x 2, and opposite side c if 1 x 2.

Determining the Largest Angle

The cosine of the largest angle A can be calculated using the Law of Cosines:

cos A  frac{b^2   c^2 - a^2}{2bc}

Substituting the values of a, b, and c, we get:

cos A  frac{(2x - 1)^2   (x^2 - 1)^2 - (x^2 - x   1)^2}{2(2x - 1)(x^2 - 1)}

Simplifying this expression, we find:

cos A  frac{-2x - 1   x - 1}{2(2x - 1)(x^2 - 1)}  -frac{1}{2}

This indicates that A 120^circ.

Final Answer

The largest angle of the triangle is 120^circ.