Problem 50 - A Graphical Approach to Precalculus with Limits

Question

CONCEPT CHECK related to the geometric property that the sum of the lengths of any two sides of a triangle must be greater than the remaining side?

Step-by-Step Solution

Answer

Use the Triangle Inequality Theorem to determine if side lengths form a triangle.

Step 1: Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be greater than the length of the third side. This gives us three possible inequalities: \(a + b > c\), \(a + c > b\), and \(b + c > a\).

Step 2: Apply the Theorem

Consider a triangle with sides of lengths \(a\), \(b\), and \(c\). To apply the Triangle Inequality Theorem, verify each of these inequalities: \(a + b > c\), \(a + c > b\), and \(b + c > a\). Each inequality checks that the sum of two sides is greater than the third side.

Step 3: Example Verification

Suppose you have a triangle with sides \(4\), \(5\), and \(6\). Check each inequality: \(4 + 5 > 6\), \(4 + 6 > 5\), and \(5 + 6 > 4\). - For \(4 + 5 > 6\), the inequality holds because \(9 > 6\).- For \(4 + 6 > 5\), the inequality holds because \(10 > 5\).- For \(5 + 6 > 4\), the inequality holds because \(11 > 4\). Since all inequalities hold, these lengths can form a triangle.

Key Concepts

Geometry

Geometry is a branch of mathematics that explores the sizes, shapes, and properties of figures and spaces. It's all about understanding how things fit together in the space around us. In the context of the Triangle Inequality Theorem, geometry helps us discover the relationship between the sides of a triangle.
Triangles are a fundamental shape studied in geometry. Triangles have three sides and three angles, and the sum of their interior angles always equals 180 degrees. Their properties form the basis for many geometric principles.
Understanding geometric concepts like the Triangle Inequality Theorem helps in solving problems related to different shapes. By mastering geometry, you can gain insights into how various shapes relate to each other and how to solve problems involving spatial reasoning.

Triangles

Triangles are polygons with three sides and three vertices. They come in different types, such as:

• Scalene triangles: All sides of different lengths.
• Isosceles triangles: Two sides of equal length.
• Equilateral triangles: All three sides of equal length.

Triangles are not only defined by their sides but also by their angles. There are three primary attributes related to angles:

• Acute triangles: All angles less than 90 degrees.
• Right triangles: One angle is exactly 90 degrees.
• Obtuse triangles: One angle greater than 90 degrees.

Triangles form the foundation of many geometric principles and theorems, such as the Triangle Inequality Theorem. This theorem ensures that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This principle helps check if given side lengths can form a valid triangle.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, often indicating that one is larger or smaller than the other. In the case of triangles, inequalities can help us understand the constraints on the side lengths.
The Triangle Inequality Theorem is a perfect example of how inequalities are used in geometry. According to this theorem:

• The sum of any two sides must be greater than the third side: \(a + b > c\)
• The sum of any two sides must be greater than the third side: \(a + c > b\)
• The sum of any two sides must be greater than the third side: \(b + c > a\)

These inequalities ensure that the sides can connect to form a triangle, much like puzzle pieces fitting together. By verifying these conditions, you can confirm whether a set of side lengths can represent a real triangle. This application of inequalities has practical importance in fields like architecture and engineering, where constructing stable structures is crucial.