Problem 42 - Prealgebra

Question

State three numbers that could be the measures of the sides of a right triangle. Justify your answer.

Step-by-Step Solution

Answer

3, 4, and 5 can be sides of a right triangle, following the Pythagorean Theorem.

Step 1: Understand the Pythagorean Theorem

For any right triangle with sides of lengths \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), the Pythagorean Theorem states that \(a^2 + b^2 = c^2\). This theorem helps us determine if a set of three numbers can be the sides of a right triangle.

Step 2: Choose a Pythagorean Triple

A Pythagorean triple is a set of three positive integers \((a, b, c)\) that work in the equation \(a^2 + b^2 = c^2\). Let's choose the well-known triple: 3, 4, and 5.

Step 3: Verify the Triangle

Substitute the numbers into the Pythagorean Theorem to verify that they form a right triangle: \(3^2 + 4^2 = 5^2\). Calculate the squares: \(9 + 16 = 25\). Since both sides of the equation are equal, these numbers can be the measures of the sides of a right triangle.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. This 90-degree angle is essential because it forms the basis for the Pythagorean Theorem. A right triangle always has three sides: the two shorter sides, known as the legs, and the longest side, called the hypotenuse. The hypotenuse is the side opposite the right angle.

In a right triangle, the relationships between the lengths of the sides are crucial for various calculations. These relationships also help us solve problems related to angles and distances. This understanding is the stepping stone to delve deeper into trigonometry and geometry concepts.

When identifying a right triangle, it's essential to check if the sum of the squares of the two legs equals the square of the hypotenuse. This property is known as the Pythagorean Theorem. It's a vital tool in determining whether a given triangle is a right triangle.

Pythagorean Triple

A Pythagorean triple consists of three positive integers that satisfy the equation derived from the Pythagorean Theorem: \(a^2 + b^2 = c^2\). This equation underlines the relationship between the sides of a right triangle, where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. Pythagorean triples are especially powerful because they provide a simple way to prove a set of numbers can form the sides of a right triangle.

Some commonly known Pythagorean triples include:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17

These sets of numbers can help in easily constructing right triangles and can even be scaled by multiplying each of the three numbers by the same integer to generate additional, larger triples. Understanding these triples is not just beneficial in academic settings, but also in practical applications where constructions and measurements are key.

Triangle Sides

Every triangle, including right triangles, consists of three sides. For right triangles specifically, these sides maintain a particular relationship due to the presence of a right angle. Here are some important aspects about the sides of triangles:

• **The Hypotenuse**: This is the longest side of a right triangle and is opposite the right angle.
• **The Legs**: These are the two other sides of the triangle. In geometric problems, these sides are often the dimensions that participate directly in the application of the Pythagorean Theorem.
• **Proportionality**: The sides of a right triangle follow a specific ratio dictated by the Pythagorean Theorem. If you have the length of one side and the hypotenuse, you can determine the length of the other side using algebraic rearrangements.

Understanding the properties and the relationships of triangle sides can make solving related problems intuitive and less time-consuming. This comprehension enables one to apply the right mathematical tools, like the Pythagorean Theorem, to figure out unknown side lengths when tasked with real-world problems or geometric puzzles.