Problem 8 - Elementary and Intermediate Algebra

Question

The shorter leg of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is ____ as long as the hypotenuse.

Step-by-Step Solution

Answer

The shorter leg is half as long as the hypotenuse.

Step 1: Understanding the Triangle

The 30-60-90 triangle is a special type of right triangle where the angles are 30°, 60°, and 90°. It has sides in a specific ratio: 1:√3:2. The shortest leg is opposite the 30° angle.

Step 2: Identifying the Hypotenuse

In a 30-60-90 triangle, the hypotenuse is always twice as long as the shortest leg, which is opposite the 30° angle.

Step 3: Relating the Shorter Leg to the Hypotenuse

Given the ratio 1:√3:2, we know the shorter leg (the one opposite 30°) is exactly half the length of the hypotenuse. Therefore, if the hypotenuse is denoted as 2x, the shorter leg is x.

Key Concepts

Understanding the 30-60-90 Triangle

The 30-60-90 triangle is a type of right triangle characterized by its three angles: 30 degrees, 60 degrees, and 90 degrees. This special triangle is known for its consistent side ratio properties, serving as a powerful tool in geometry.

What makes this triangle unique is its predictable ratio of side lengths. In a 30-60-90 triangle, the ratios of the sides are always 1:√3:2. Each side of the triangle is related to its corresponding angle in a set pattern. The side opposite the 30-degree angle is the shortest. This structure helps us easily calculate side lengths without complex measurements when at least one side length is known.

Knowing this, you can quickly determine the measurements needed for solving geometry problems involving these triangles, which is a common occurrence in various math exercises.

Triangle Side Ratios Demystified

Understanding triangle side ratios can greatly simplify solving geometry problems, especially with 30-60-90 triangles. In such triangles, the specific ratio—1:√3:2—describes the relative lengths of each side.

• The shortest side: Always opposite the 30-degree angle, and is effectively the base ratio of 1. This means if you know the length of this side, you can determine the others.
• Middle side: This is opposite the 60-degree angle and is √3 times the length of the shortest side. Finding this side requires basic multiplication using the known shortest side.
• The hypotenuse: Always opposite the 90-degree angle and is twice the length of the shortest side. This attribute is critical for confirming calculations and checking your work in mathematical problems.

The clarity and predictability of these ratios mean you can approach problems with confidence, knowing you're working from established geometric principles.

Special Triangles and Their Importance

Special triangles, such as the 30-60-90 triangle, play a key role in simplifying complex geometry problems and are often pivotal elements in math education. They serve as a shortcut for students and mathematicians, with predictable patterns making calculations easier.

A special triangle like the 30-60-90 provides instant understanding of side relationships without needing deeper trigonometric functions each time. This is immensely helpful when dealing with repeated calculations or when shortcuts are beneficial.

These triangles are frequently used not just in classroom settings but also in different fields such as architecture, engineering, and even in computer graphics, due to their consistent properties. Understanding how they work enhances both foundational mathematics skills and practical applications in real-world scenarios.