Problem 8 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. If the length of the hypotenuse of a right triangle is \(c\) and the lengths of the other two legs are \(a\) and \(b\), then \(=c^{2}\) b. In a right triangle, the sum of the ____________ of the lengths of the two legs is equal to the square of the length of the ______

Step-by-Step Solution

Answer

a. \(c^2 = a^2 + b^2\); b. squares, hypotenuse.

Step 1: Understanding the Pythagorean Theorem

In a right triangle, the Pythagorean Theorem states that the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two legs (\(a\) and \(b\)). Mathematically, it is expressed as \(a^2 + b^2 = c^2\).

Step 2: Filling in Part (a)

For part (a), we need to express the square of the hypotenuse in terms of the squares of the other two legs. According to the Pythagorean Theorem, \(c^2 = a^2 + b^2\).

Step 3: Filling in Part (b)

In a right triangle, the phrase 'the sum of the ____________ of the lengths of the two legs' refers to the sum of the squares of the lengths of the two legs. The next blank refers to the hypotenuse, so the complete sentence is: 'the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.'

Key Concepts

Understanding the Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is called a right angle. The side opposite this angle is the longest side and is unique to right triangles. This side is known as the hypotenuse. The other two sides that form the right angle are called the legs of the triangle. Together, they often help determine the properties of the triangle using various mathematical principles like the Pythagorean Theorem. Understanding the right triangle is a fundamental part of geometry that allows students to explore the relationships between different shapes and lengths. It is important to identify each side of a right triangle correctly in order to use mathematical formulas effectively.

Exploring the Hypotenuse

The hypotenuse is the side of a right triangle opposite the right angle. It is always the longest side because it is opposite the largest angle. The concept of the hypotenuse is central to the Pythagorean Theorem, which helps us find unknown sides in right triangles. When you know the lengths of the two other sides (the legs), you can calculate the hypotenuse using the theorem. Here is a little guide on how to identify and use the hypotenuse in your calculations:

• The hypotenuse is opposite the right angle.
• It is the side you solve for in equations using the Pythagorean Theorem.
• It is typically labeled as "c" in mathematical equations and problems.

Remember, in a right triangle, the hypotenuse will always be longer than either of the other two sides.

The Concept of Sum of Squares

The term 'sum of squares' is a crucial part of the Pythagorean Theorem. It refers to the addition of the squares of a triangle's two leg lengths. Mathematically, it can be written as \(a^2 + b^2\). Each leg (lengths \(a\) and \(b\)) is squared, and their results are summed. The sum of these squares provides you with the square of the hypotenuse. This equation is expressed as \(a^2 + b^2 = c^2\). Here is a simple breakdown of the concept:

• Square each leg first: find \(a^2\) and \(b^2\).
• Add them together: \(a^2 + b^2\).
• Set equal to the square of the hypotenuse: equals \(c^2\).

This formula helps in various practical applications, such as determining the unknown lengths of a triangle's sides, ensuring accurate construction measurements, or even in navigation when calculating direct paths.