Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Without using the law of sines, explain why no triangle \(A B C\) exists satisfying \(A=103^{\circ} 20^{\prime}, a=14.6\) feet, and \(b=20.4\) feet.

Step-by-Step Solution

Answer

No triangle exists because side opposite largest angle is not the largest.

Step 1: Recall the Triangle Inequality Theorem

According to the Triangle Inequality Theorem, for any triangle with sides of length \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be greater than the length of the third side. Mathematically, this means that \(a + b > c\), \(a + c > b\), and \(b + c > a\).

Step 2: Understanding the Given Angles and Sides

We are given that \(A = 103^{\circ} 20^\prime\), \(a = 14.6\) feet, and \(b = 20.4\) feet. In triangle \(ABC\), if \(A\) is the largest angle, the side opposite to it, \(a\), should be the largest side.

Step 3: Analyze the Largest Angle Condition

Since angle \(A\) is the largest at \(103^{\circ} 20^\prime\), side \(a\) opposite to angle \(A\) should be larger than side \(b\), because larger angles in a triangle subtend larger opposite sides. However, the given \(a = 14.6\) feet is less than \(b = 20.4\) feet. This is a contradiction.

Step 4: Conclude That No Such Triangle Exists

Given that angle \(A\) is the largest, but its opposite side is not the largest compared to \(b\), the condition contradicts the properties of a triangle. Therefore, no triangle \(ABC\) can exist under these conditions.

Key Concepts

Geometry

In the study of geometry, one of the fundamental shapes we encounter is the triangle. Each triangle has three sides and three angles, and its properties are governed by specific rules and theorems.
A triangle can be classified based on its angles or sides. For instance, based on angles, we can have acute, obtuse, and right triangles. Based on sides, there are equilateral, isosceles, and scalene triangles.
A critical aspect of understanding triangles is ensuring all properties align with these definitions and theorems. This is where the Triangle Inequality Theorem, among others, plays a crucial role. This theorem helps us determine if a set of three lengths can form a triangle.

Triangle Properties

The properties of a triangle provide valuable insights into its structure and relationships between its components. A vital set of properties concerns the relationship between the sides and angles.
Some important properties include:

• The sum of all internal angles of a triangle is always 180 degrees.
• The longest side of a triangle is opposite the largest angle.
• The shortest side of a triangle is opposite the smallest angle.
• Each side is less than the sum of the other two sides but greater than the difference between them, as detailed by the Triangle Inequality Theorem.

These properties ensure that all components of a triangle harmonize, which is essential when validating the existence of a triangle based on given side lengths and angles.

Angle-Side Relationship

The relationship between the angles and sides of a triangle is fundamental to understanding its geometry. This relationship dictates how the sides interact based on the angles within the triangle.
According to this principle, the size of an angle within a triangle directly affects the length of the side opposite to it. Therefore, larger angles are opposite longer sides, and smaller angles are opposite shorter sides. This is often described as the "angle-side relationship".
For example, in the problem presented, the largest angle was given as \(103^{\circ} 20'\), which should typically relate to the longest side. However, if the opposite side is not the longest, as shown in the problem, it results in a contradiction, revealing that no such triangle can exist. This clear linkage between angles and sides helps in verifying the feasibility of a triangle's existence.