Problem 9 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The length of the longer leg of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is _____ times the length of the shorter leg.

Step-by-Step Solution

Answer

\(\sqrt{3}\)

Step 1: Understand the characteristics of the triangle

A \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is a special type of right triangle with specific side ratio properties. The sides opposite the \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) angles are in the ratio \(1 : \sqrt{3} : 2\) respectively.

Step 2: Identify the relationship between the shorter and longer leg

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the shorter leg is opposite the \(30^{\circ}\) angle, and the longer leg is opposite the \(60^{\circ}\) angle. According to the side ratios \(1 : \sqrt{3}\), the longer leg is \(\sqrt{3}\) times the shorter leg.

Key Concepts

30-60-90 Triangle

A 30-60-90 triangle is a special kind of right triangle that occurs frequently in geometry. It is called a 30-60-90 triangle because the measures of its angles are 30 degrees, 60 degrees, and 90 degrees. This type of triangle is important because it has a unique set of side length ratios, making it predictable and easy to work with.

This kind of triangle is often found in problems related to trigonometry and geometry, and understanding its properties can make solving these problems much simpler. Given its fixed angle measurements, students can apply the known side ratios to find missing side lengths or quickly verify calculations.

Side Ratios

When dealing with a 30-60-90 triangle, it's essential to understand the specific side ratios. The sides opposite the 30-degree, 60-degree, and 90-degree angles have a fixed ratio of 1 : \( \sqrt{3} \) : 2. This means:

• The side opposite the 30-degree angle (shorter leg) is the smallest, and is labeled as 1 unit in our ratio.
• The side opposite the 60-degree angle (longer leg) has a length of \( \sqrt{3} \) times the shorter leg.
• The side opposite the 90-degree angle (hypotenuse) is twice as long as the shorter leg.

These ratios allow you to calculate the length of any side if at least one side length is known, by simply applying multiplication based on these ratio values. The longest side, the hypotenuse, will always be twice the length of the shorter leg, and the longer leg is \( \sqrt{3} \) times the length of the shorter leg.

Right Triangle Properties

Right triangles have unique properties that distinguish them from other triangle types. They have one 90-degree angle, which makes calculations simpler by letting the Pythagorean theorem be used. In a right triangle like the 30-60-90 triangle, these properties help solve for unknown side lengths using both angles and their relationships to side lengths.

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. However, in a 30-60-90 triangle, knowing one side and the unique side ratios also provides a quick and direct way to find missing side lengths without always resorting to the theorem.