Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. In a right triangle, the side opposite the \(90^{\circ}\) angle is called the _______.

Step-by-Step Solution

Answer

The side is called the hypotenuse.

Step 1: Identify Triangle Types

A triangle with one angle measuring 90 degrees is known as a right triangle. This is because it includes a right angle, which is exactly 90 degrees.

Step 2: Understand Triangle Parts

In a right triangle, there are three sides, each with a specific name depending on their position relative to the right angle. The side directly opposite the 90-degree angle is always the longest side.

Step 3: Identify the Longest Side

In the context of a right triangle, the longest side which is opposite the 90-degree angle is referred to as the hypotenuse.

Step 4: Fill in the Blank

Based on the information identified in the previous steps, fill in the blank with 'hypotenuse' for the name of the side opposite the 90-degree angle.

Key Concepts

Hypotenuse

Every right triangle, which is a triangle featuring one of its angles as exactly 90 degrees, has a special side known as the hypotenuse. This is always the longest side of the right triangle. Think of the hypotenuse as the backbone or the main side of the triangle. It stretches out on the opposite side of the right angle itself.

The hypotenuse holds great importance because it appears in the Pythagorean theorem, a vital topic in trigonometry and geometry for solving real-world problems. This theorem asserts that the square of the hypotenuse, denoted as \(c\), equals the sum of the squares of the other two sides, known as \(a\) and \(b\). Mathematically, this can be expressed as:

• \(c^2 = a^2 + b^2\)

This formula is useful across diverse fields such as construction, navigation, and even computer graphics, making the hypotenuse a key element in correct triangle calculations.

Right Angle

The right angle is one of the most straightforward yet powerful concepts in geometry. In a right triangle, one of the angles is precisely 90 degrees. This right angle is what gives these types of triangles their name.

The right angle acts like a pillar in the triangle. It not only defines the shape of the triangle but also classifies it distinctly from other types of triangles like acute or obtuse triangles. A right angle looks like the corner of a typical square or rectangle and often makes right triangles useful in practical design and architecture.

When you spot a right angle in any geometric figure, you can determine that it's part of a right triangle, or potentially forms a part of a rectangle or square. Right angles make calculations more manageable because they allow for straightforward application of the Pythagorean theorem and trigonometric ratios like sine, cosine, and tangent.

Triangle Sides

In understanding right triangles, it’s essential to familiarize yourself with the three sides, each playing a vital role. These sides include the hypotenuse and the two legs.

• Hypotenuse: As previously discussed, this is the longest side, opposite the right angle.
• Legs: The two sides forming the right angle are known as the legs of the right triangle. They are shorter than the hypotenuse and are crucial for defining the triangle's height and base.

The relationship between the sides is beautifully showcased in the Pythagorean theorem, which helps in calculating the length of one side if the lengths of the other two are known. Each of these sides can also help determine other trigonometric functions, aiding in deeper analysis of angles and shapes within the right triangle. This clarity in understanding triangle sides simplifies many complex geometrical and practical problems.