Question

Critical Thinking In an obtuse triangle, why is the longest side opposite the obtuse angle?

Short answer

The longest side is opposite the obtuse angle because it is the largest angle in the triangle.

Step-by-step solution

Understand the Definition of an Obtuse Triangle

An obtuse triangle is a triangle where one of its interior angles measures more than 90 degrees. This angle is called the obtuse angle. The other two angles are necessarily less than 90 degrees each.

Analyze the Relationship Between Angles and Opposite Sides in a Triangle

In any triangle, the side opposite the largest angle is always the longest side. This is due to the Law of Sines and the basic properties of triangles, where larger angles have longer opposite sides.

Apply the Relationship to an Obtuse Triangle

In an obtuse triangle, the obtuse angle is the largest angle because it is greater than 90 degrees. According to the relationship examined in Step 2, the side opposite this largest angle will be the longest side.

Conclude the Reasoning

Since the longest side is opposite the largest angle in any triangle, and the obtuse angle is the largest angle in an obtuse triangle, it logically follows that the longest side must be opposite the obtuse angle.

Key concepts

Angle-Side Relationship in Triangles

The angle-side relationship in triangles is a fundamental principle that helps us understand how the sizes of the angles in a triangle affect the lengths of the sides. In simpler terms, the bigger the angle, the longer the side opposite to it. This relationship is based on the geometric properties of triangles and holds true for all types of triangles, including obtuse triangles.
Here’s why:

• Each angle in a triangle is associated with the side opposite it.
• The largest angle will have the largest opposite side due to the expansive reach of the angle.
• This rule does not change whether the triangle is acute, right, or obtuse.

In the context of an obtuse triangle, this principle is crucial. An obtuse angle, by definition, measures more than 90 degrees, making it the largest angle in that triangle. Thus, the side facing this obtuse angle will necessarily be the longest. This understanding aids in visualizing and solving problems related to triangle geometry.

Understanding the Law of Sines

The Law of Sines is an essential tool in trigonometry for exploring relationships between the angles and sides of a triangle. It is particularly useful when dealing with non-right triangles, including obtuse ones.
The Law of Sines states:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]where \(a, b, c\) are the side lengths of a triangle, and \(A, B, C\) are the corresponding opposite angles.
This formula shows how each side length is directly proportional to the sine of its opposite angle. Consequently:

• When the angle is larger, like an obtuse angle, the sine value is greater than that of an acute angle, resulting in a proportionally longer side length.
• The longest side in any triangle will always align with the largest angle due to this proportional relationship.

Applying the Law of Sines in an obtuse triangle thus shows why the side opposite the obtuse angle must be the longest, reiterating the angle-side relationship.

Core Properties of Triangles

Understanding the properties of triangles is key in solving geometric problems involving triangles, including obtuse triangles.
Some basic properties include:

• Triangles always have a total internal angle sum of 180 degrees.
• The angles determine the triangle type: acute, right, or obtuse.
• An obtuse triangle has one angle greater than 90 degrees, making it the defining feature of this triangle type.
• In any triangle, the sum of any two sides must be greater than the third side (Triangle Inequality Theorem).

Specific to obtuse triangles:

• The presence of an angle larger than 90 degrees impacts the potential triangle configurations.
• These configurations dictate that the longest side will always complement the obtuse angle, adhering to the angle-side relationship.
• Thus, internal geometric rules ensure consistency across different triangle types.

These properties and principles ensure predictability in solving triangle-related problems, providing a solid foundation for understanding more complex geometric concepts.