Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The longest side of a right triangle is the _____________ . The remaining two sides are the _______________ of the triangle.

Step-by-Step Solution

Answer

The hypotenuse; the legs.

Step 1: Identify Right Triangle Components

A right triangle has one angle exactly equal to 90 degrees. In such a triangle, the side opposite the right angle is the longest side.

Step 2: Name the Longest Side

The longest side of a right triangle is known as the hypotenuse. It is the side that is always opposite the right angle.

Step 3: Name the Other Two Sides

The remaining two sides of the right triangle are called the legs, or sometimes referred to as the catheti (singular: cathetus). These sides form the right angle in the triangle.

Key Concepts

Hypotenuse

In a right triangle, the hypotenuse is a critical component. This is because it is the longest side of the triangle. The unique aspect of a right triangle is that it has one angle measuring exactly 90 degrees - this is known as the right angle. The hypotenuse is always opposite this right angle, serving as a sort of bridge or diagonal that connects the two non-right angled sides.
A great way to remember the hypotenuse is to think of it as the side that stretches the longest across the triangle's peak, opposite the right angle. This makes it easy to spot in diagrams.

• The hypotenuse is crucial in calculations, especially in the Pythagorean Theorem.
• It helps calculate distances and measurements in practical applications like construction and navigation.

The equation used in the Pythagorean Theorem is:
\[ c^2 = a^2 + b^2 \]
where \( c \) represents the hypotenuse.

Legs of a Triangle

In a right triangle, the legs are the two sides that together form the right angle. These two sides are important as they determine the dimensions and shape of the triangle alongside the hypotenuse.
The legs are shorter than the hypotenuse, and they meet at the right angle. They are essential in understanding how right triangles function because, without them, the triangle would not have its signature 90-degree angle.

• They are called 'legs' because they appear as the base and height of a triangle in diagrams.
• The legs can be equal or different lengths, depending on whether the triangle is an isosceles right triangle or a scalene right triangle.
• These sides are involved in the calculation of area: \[ \text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 \]

Whether they're considered the base or height, the legs are integral to the right triangle's structure and applications.

Right Angle

The right angle is the defining feature of a right triangle. What makes this angle special is that it measures exactly 90 degrees, creating the distinct L-shape many recognize in right triangles. The presence of a right angle is what transforms an ordinary triangle into a right triangle and it provides a firm reference point for identifying other triangle components.
When a right angle is present, the triangle's structure allows for the use of many mathematical formulas, especially the Pythagorean Theorem, which applies only to right triangles.

• Right angles are characterized by their presence in many everyday objects. For example, corners of paper and squares.
• They are integral in architecture and engineering, providing necessary stability.
• In trigonometry, right angles help in the definitions and calculations of sine, cosine, and tangent.

Thus, the right angle is not only the essential angle in a right triangle but also a foundational element in many areas of mathematics and real-world applications.