Problem 36 - Algebra and Trigonometry

Question

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=4, \quad c=5, \quad B=95^{\circ} $$

Step-by-Step Solution

Answer

No triangle can be formed.

Step 1: Analyze the given information

The sides and an angle given are: \(b = 4\)\(c = 5\)\(B = 95^{\circ}\) Since we need to determine how many triangles can be formed and solve them, we must first understand the relationship between these components.

Step 2: Check the type of triangle

We know that angle B is more than 90 degrees, making it an obtuse angle. In any triangle where an obtuse angle is given, the side opposite this angle will be the longest side. Here, since angle B is 95 degrees, side b should be longer than side c.

Step 3: Determine feasibility

Given that side b (4) is shorter than side c (5), this contradicts the property of triangles containing an obtuse angle, where the side opposite the obtuse angle must be the longest side. Therefore, no triangle can be formed with the given parameters.

Key Concepts

Triangle Properties

Understanding the fundamental properties of triangles is essential when solving triangle-related problems. A triangle is a three-sided polygon, and it has several important properties that define its shape and angles. Here are a few key properties every student should know:

• Each triangle has three sides and three angles.
• The sum of the internal angles of any triangle always equals 180 degrees.
• Triangles can be classified by their angles (acute, right, obtuse) or by their sides (equilateral, isosceles, scalene).

These properties help determine the feasibility of forming a triangle with given sides and angles. In our exercise, identifying that angle B is obtuse influences the relationship between the sides and the angle.

Obtuse Angle

An obtuse angle is one that measures more than 90 degrees but less than 180 degrees. In any triangle containing an obtuse angle, several properties become important:

• The side opposite the obtuse angle is always the longest side of the triangle.
• Given an obtuse angle, the other two angles must be acute (each less than 90 degrees) since the total must sum to 180 degrees.

In our exercise, angle B is 95 degrees, making it an obtuse angle. According to the properties of obtuse angles, side b should be the longest side. However, since side b (4) is shorter than side c (5), this creates a contradiction. Therefore, it's impossible to form a triangle with the given sides and obtuse angle.

Triangle Inequality Theorem

The Triangle Inequality Theorem is crucial in determining whether a set of three sides can form a triangle. It states that:
x+y>z \ \ \,\text{(and also:)} \[x < y + z, \y < x + z, \z < x + y\]where x, y, and z are the lengths of the sides of a triangle. This theorem must be satisfied for all three combinations of side lengths in any triangle.
Applying this theorem to our exercise (though our missing side length isn't directly given), we would set constraints based on valid triangle conditions. However, due to the obtuseness of angle B and the corresponding required length of side b, the combination of sides and angles given does not create a valid triangle under these principles. Thus, ensuring that given sides and angles adhere to the Triangle Inequality Theorem and properties of triangle types is essential for solving triangle problems accurately.