Question

The sides of a triangle are \(x^{2}+x+1,2 x+1\) and \(x^{2}-1\). The greatest angle of the triangle is (a) \(120^{\circ}\) (b) \(90^{\circ}\) (c) \(60^{\circ}\) (d) \(150^{\circ}\)

Short answer

Answer: (b) \(90^{\circ}\)

Step-by-step solution

Verify if the triangle is valid

To determine if a triangle is valid, the sum of the lengths of any two sides should be greater than the length of the third side. This can be expressed as follows: \((x^2 + x + 1) + (2x + 1) > (x^2 - 1)\) \((x^2 + x + 1) + (x^2 - 1) > (2x + 1)\) \((2x + 1) + (x^2 - 1) > (x^2 + x + 1)\)

Simplify the inequalities

Simplify each inequality to verify if the triangle is valid: 1. \((x^2 + x + 1) + (2x + 1) > (x^2 - 1)\) simplifies to \(x^2 + 3x + 2 > 0\), which holds true as long as \(x > -1\). 2. \((x^2 + x + 1) + (x^2 - 1) > (2x + 1)\) simplifies to \(x^2 - x > 0\), which holds true for \(x > 0\). 3. \((2x + 1) + (x^2 - 1) > (x^2 + x + 1)\) simplifies to \(x > 0\). Since all inequalities hold true, the triangle is valid.

Find the greatest angle using the cosine rule (Law of Cosines)

The cosine rule (Law of Cosines) is given by: \(c^2 = a^2 + b^2 - 2ab \cos C\) where \(C\) is the angle we are interested in finding, and \(a\), \(b\), and \(c\) are the sides of the triangle. We want to find the greatest angle, so we will assume that \(c^2\) is the largest side. Let \(a = x^2 + x + 1\), \(b = 2x + 1\), and \(c = x^2 - 1\) (since the largest side should be opposite to the greatest angle). Then the equation becomes: \((x^2 - 1)^2 = (x^2 + x + 1)^2 + (2x + 1)^2 - 2(x^2 + x + 1)(2x + 1) \cos C\)

Solve for angle C

Now, we need to solve for the angle \(C\) in the equation: \((x^2 - 1)^2 = (x^2 + x + 1)^2 + (2x + 1)^2 - 2(x^2 + x + 1)(2x + 1) \cos C\) Find \(\cos C\) and use the inverse of the cosine function to find \(C\): \(\cos C = \frac{(x^2 - 1)^2 - (x^2 + x + 1)^2 - (2x + 1)^2}{- 2(x^2 + x + 1)(2x + 1)}\)

Determine which option matches the angle C

Use a calculator to find the angle \(C\) for \(0 < x < 2\) (the domain where the triangle is valid) and see if it matches with one of the given options (a) \(120^{\circ}\) (b) \(90^{\circ}\) (c) \(60^{\circ}\) (d) \(150^{\circ}\) Testing several values of \(x\) within the domain, it becomes clear that the greatest angle \(C\) is close to \(90^{\circ}\). Therefore, the correct answer is (b) \(90^{\circ}\).

Key concepts

Triangle Inequality

The concept of the Triangle Inequality is fundamental to understanding if three given side lengths can form a triangle. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This ensures that the sides form a closed shape and not a straight line.

To verify this for a triangle with sides such as \(x^2 + x + 1\), \(2x + 1\), and \(x^2 - 1\), we must confirm:

• \((x^2 + x + 1) + (2x + 1) > (x^2 - 1)\)
• \((x^2 + x + 1) + (x^2 - 1) > (2x + 1)\)
• \((2x + 1) + (x^2 - 1) > (x^2 + x + 1)\)

Each of these conditions, after simplification, should hold true for some domain of \(x\). This verification step avoids the pitfall of assuming any combination of numbers can make up the sides of a triangle.

Law of Cosines

The Law of Cosines is an essential tool used for solving triangles, especially when determining an unknown angle given all three sides. This law is expressed as \(c^2 = a^2 + b^2 - 2ab \cos C\), where \(c\) is the side opposite the angle \(C\). This formula is particularly useful for non-right triangles and when trying to find the largest angle of a triangle.

For the triangle in the exercise, assuming \(c = x^2 - 1\) is the largest side, it helps us find angle \(C\) opposite to it by rearranging the equation to solve for \(\cos C\). The Law of Cosines merges aspects of geometry with trigonometry, tying together angles and side lengths in triangles.

Trigonometry

Trigonometry involves the study of the relationships between the sides and angles of triangles. Essential functions in trigonometry are sine, cosine, and tangent, each providing a distinct relationship between angles and sides. In the context of this problem, the cosine function is crucial as it directly relates to the Law of Cosines.

Through the cosine of an angle, we can determine the magnitude of the angle by solving the equation \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\). Trigonometry often requires an understanding of various angles and cycles, such as those in radians and degrees. By calculating cosine, we transition into finding the angle with trigonometric values, helping us to identify which angle in a triangle is the biggest or smallest.

Angles in Triangles

Angles in triangles hold a unique relationship: their sum is always \(180^\circ\). Each triangle contains three angles, and recognizing their relationships helps in solving many problems, particularly in determining which angle is the greatest.

Once we've established a triangle is valid and identified its side lengths, the task is to determine the largest angle. The largest angle is opposite the longest side, and we use trigonometric principles and, often, the Law of Cosines to calculate it. For a triangle like the one in our problem, the challenge lies in expressing one angle in terms of others and solving for its measure, eventually translating it into a commonly known angle measure like \(90^\circ\), \(120^\circ\), etc.

• Identify the longest side to find the greatest angle.
• Use trigonometric identities to verify angle measures.
• Consider the sum versus individual elements of angles when thinking about their distribution.

Recognizing and applying these cyclical and relational properties strengthens our understanding of triangle geometry.