In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 3, b = 3, c = 3

Verified step by step guidance

1. insert step 1: Recognize that the given triangle is an equilateral triangle because all sides are equal (a = b = c = 3).
2. insert step 2: In an equilateral triangle, all angles are equal. Therefore, each angle measures 60 degrees.
3. insert step 3: Use the Law of Cosines to verify the angle measures. The Law of Cosines states: c^2 = a^2 + b^2 - 2ab \(\cdot\) \(\cos\)(C).
4. insert step 4: Substitute the known values into the Law of Cosines: 3^2 = 3^2 + 3^2 - 2 \(\cdot\) 3 \(\cdot\) 3 \(\cdot\) \(\cos\)(C).
5. insert step 5: Simplify the equation to find \(\cos\)(C) and confirm that C = 60 degrees.

Key Concepts

Triangle Properties

Understanding the properties of triangles is essential for solving them. A triangle consists of three sides and three angles, and the sum of the interior angles always equals 180 degrees. In this case, since all sides are equal (a = b = c), the triangle is equilateral, meaning all angles are also equal, each measuring 60 degrees.

Law of Cosines

The Law of Cosines is a crucial formula used to find unknown sides or angles in any triangle. It states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the relationship is given by c² = a² + b² - 2ab * cos(C). This law is particularly useful when dealing with non-right triangles, as it allows for the calculation of angles and sides based on known values.

Rounding and Precision

Rounding is an important aspect of presenting numerical answers in a clear and concise manner. In this exercise, lengths are to be rounded to the nearest tenth and angles to the nearest degree. This means that when calculating the lengths of sides or measures of angles, one must apply rounding rules to ensure the final answers are appropriately formatted, which is crucial for accuracy in practical applications.