Problem 8 - Intermediate Algebra

Question

Fill in the blanks. 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

In a special right triangle, such as the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, there are specific ratio relationships between the lengths of the sides. The sides are in the ratio \(1:\sqrt{3}:2\).

Step 2: Identifying the Shorter Leg

The shorter leg is the side opposite the \(30^{\circ}\) angle in a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, and this leg corresponds to the value \(1\) in the ratio \(1:\sqrt{3}:2\).

Step 3: Analyzing the Ratio to the Hypotenuse

In the ratio \(1:\sqrt{3}:2\), the hypotenuse is \(2\). Thus, the shorter leg is \(\frac{1}{2}\) as long as the hypotenuse.

Key Concepts

30-60-90 triangle

The 30-60-90 triangle is a special kind of right triangle. What's unique about it? The angles of this triangle are always 30 degrees, 60 degrees, and 90 degrees. Because of these angles, the triangle has specific properties and predictable side lengths. It's like a blueprint for solving many math problems. These predictable patterns come in handy, especially in geometry, trigonometry, and problems involving right triangles.

• One angle is always a right angle, which measures 90 degrees.
• The other two angles are 30 degrees and 60 degrees, adding up to make the total three angles equal to 180 degrees, as in any triangle.

Knowing these angles helps students quickly determine properties of the triangle's sides and solve related mathematical problems efficiently.

side length ratios

In the 30-60-90 triangle, the side lengths follow a special ratio. These ratios are a result of the specific angles inside the triangle and are always consistent. You can think of these ratios as rules that never change, much like a recipe.

• The side opposite the 30-degree angle is the shortest and is taken as 1 part.
• The side opposite the 60-degree angle is longer, specifically \(\sqrt{3}\) times the shortest side.
• The hypotenuse, opposite the right angle, is the longest side, always twice the length of the shortest side.

This ratio means, if you know the length of just one side, you can easily find the others using the ratio 1 : \(\sqrt{3}\) : 2. For instance, if the shortest side (opposite the 30-degree angle) is 5 units, then the other sides are 5 \(\sqrt{3}\) and 10 units long.

special right triangles

Special right triangles, like the 30-60-90 triangle, follow predictable side length patterns that make solving problems easier. You might wonder why these triangles are called special. The answer lies in their ability to simplify complex calculations.

• Special right triangles are those that have angles of note, like 30°, 45°, or 60°.
• They reduce the need for trigonometric calculations, often using simple arithmetic instead.
• Their fixed side ratios allow for quick solutions without lengthy computations.

These triangles are widely used in geometry and trigonometry. By recognizing them, you can quickly determine unknown side lengths or angles, saving time and effort. They fundamentally help in understanding various math concepts and are a stepping-stone to grasping more complex topics in mathematics.