Problem 6 - Intermediate Algebra

Question

Fill in the blanks. In any right triangle, the square of the hypotenuse is equal to the _____ of the squares of the two _____.

Step-by-Step Solution

Answer

sum, legs

Step 1: Identify the Theorem

Recognize that the exercise refers to the Pythagorean Theorem, which applies to right triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Step 2: Fill in the First Blank: Mathematical Term

The term that describes the operation we perform with the squares of the two sides is 'sum'. Thus, the first blank is filled with the word 'sum'.

Step 3: Fill in the Second Blank: Identify the Sides

The word that describes the sides in a right triangle that are not the hypotenuse is 'legs'. Therefore, the second blank should be filled with 'legs'.

Key Concepts

Understanding Right Triangles

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This property gives the triangle its name, as the 90-degree angle is also called a "right angle." A right triangle has three sides:

• The hypotenuse
• The two shorter sides known as legs

One of the defining features of right triangles is how the Pythagorean Theorem relates their sides. Due to the properties of right angles, the relationship can be explored through the equation, which is fundamental for solving various geometry problems. The right triangle serves as the basis for many concepts in trigonometry and geometry, making it an essential component of mathematics.
It's helpful to visualize a right triangle by imagining a triangle cornered by a square or rectangle, as the right angle naturally forms where the square's edges meet.

The Hypotenuse Simplified

The hypotenuse is the longest side of a right triangle. It lies directly opposite the right angle and is crucial when applying the Pythagorean Theorem. The term "hypotenuse" might sound complex, but it's simply the side that stretches the furthest in a right triangle.
This side is significant because it forms the basis for the famous Pythagorean equation: \[c^{2} = a^{2} + b^{2}\]Here, \(c\) represents the length of the hypotenuse, while \(a\) and \(b\) are the lengths of the other two sides, known as the legs. This relationship allows you to determine the length of any side if the lengths of the other two sides are known.
In everyday life, knowing how to find the hypotenuse can help solve practical problems, such as determining the ladder length required to safely reach a certain height.

Exploring the Legs of a Right Triangle

The two shorter sides of a right triangle are known as the "legs." These sides meet at the right angle and are essential for calculating the hypotenuse using the Pythagorean Theorem.

• One leg is often referred to as the "base," and the other as the "height," depending on how the triangle is oriented.
• The legs are usually denoted as \(a\) and \(b\) in mathematical problems, forming part of the equation \(a^{2} + b^{2} = c^{2}\).

The lengths of these sides are crucial in various mathematical applications beyond just geometry. When examining the legs, one discovers their importance in real-world contexts, such as construction, navigation, and even art.
Using the legs and the Pythagorean Theorem, one can determine various other properties of the triangle, facilitating a deeper understanding of spatial relationships and structure.