Problem 26 - Algebra 2 and Trigonometry

Question

Use the Law of Cosines to prove that if the angle between two congruent sides of a triangle measures \(60^{\circ},\) the triangle is equilateral.

Step-by-Step Solution

Answer

The triangle is equilateral because all sides are equal, satisfying \(b = a\).

Step 1: Understanding the Given Information

We are given a triangle with two congruent sides and an included angle of \(60^{\circ}\). Let's denote the sides of the triangle as \(a, a,\) and \(b\), where \(a\) and \(a\) are the congruent sides, and the angle between them is \(60^{\circ}\). Our goal is to show that all three sides are equal, meaning the triangle is equilateral.

Step 2: Applying the Law of Cosines

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), opposite to angles \(A\), \(B\), and \(C\) respectively, \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\). In our case, apply it to find \(b\) (the side opposite the \(60^{\circ}\) angle): \[ b^2 = a^2 + a^2 - 2(a)(a) \cos(60^{\circ}) \] Since \(\cos(60^{\circ}) = \frac{1}{2}\), substitute it in:

Step 3: Simplifying the Law of Cosines Equation

Substituting \(\cos(60^{\circ}) = \frac{1}{2}\) in, we get: \[ b^2 = a^2 + a^2 - 2a^2 \cdot \frac{1}{2} \] Simplifying further: \[ b^2 = a^2 + a^2 - a^2 \] \[ b^2 = a^2 \]

Step 4: Concluding the Solution

Since \(b^2 = a^2\), it follows that \(b = a\). This means that all three sides of the triangle are equal \(a, a, b\), thus the triangle is equilateral.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle in which all three sides are the same length. This means each angle within the triangle is equal as well, measuring exactly 60 degrees each. Why is this important? In geometry, the equal side lengths and angle measures provide symmetry, making calculations and proofs simpler. Additionally, equilateral triangles are often used in problems related to symmetry and regularity due to their uniform shape:

• The equal sides contribute to uniformity in angles.
• Each internal angle perfectly measures 60 degrees, creating a regular polygon.
• Equilateral triangles serve as foundational elements in many geometric shapes and patterns.

Understanding the equilateral triangle is essential for recognizing patterns and solving complex geometric problems.

Congruent Sides

In the context of triangles, congruent sides refer to those sides that are of equal length. When we say a triangle has two congruent sides, we are often describing an isosceles triangle. However, if the angle between these two congruent sides is 60 degrees, the triangle is actually equilateral, as proven through the exercise. Here's why congruent sides are fundamental:

• They establish the concept of equality in geometric figures.
• In our exercise, the congruency between the sides is crucial in achieving an equilateral state when combined with the 60-degree angle.
• Congruent sides simplify the application of the Law of Cosines, as some terms cancel out and reduce complexity in calculations.

Congruent sides help lay the groundwork for more advanced geometric principles and are widely used in proofs to establish relationships between different parts of a figure.

Angle Measure

The measure of the angle is a key component in defining a triangle's type and properties. In this exercise, having a 60-degree angle between two congruent sides is what ultimately validates the equilateral nature of the triangle. Angle measures influence:

• The type of triangle—equilateral, isosceles, or scalene.
• Properties of symmetry and balance within the triangle.
• The relationships between the sides, which can be explored through trigonometric methods such as using the Law of Cosines.

Specifically, if any angle in a triangle is 60 degrees, it paves the way to identify patterns and draw conclusions about the overall geometric shape, particularly when combined with congruent sides or other known angles.

Cosine Function

The cosine function relates the angles and sides of triangles in trigonometry, playing a central role in applying the Law of Cosines. In this exercise, using \(\cos(60^{\circ}) = \frac{1}{2}\), helped simplify the problem and confirm the triangle as equilateral. Here’s how cosine impacts the situation:

• Provides a connection between the angle and side lengths in a triangle, so calculations are precise.
• Allows transformation of known quantities (like angle measures) into side length calculations through the Law of Cosines.
• Simplifies the equation to deduce other properties of the triangle, making problem-solving more straightforward.

The cosine function is not only crucial for triangle problems but also serves as a valuable tool in various fields, such as physics and engineering, where it models periodic phenomena.