Checking the First Set of Side Lengths

Oct 13, 2025 by talynt.org 11 views

Exploring Pythagorean Triples: A Mathematical Analysis

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this article, we will examine four sets of side lengths and determine which ones are Pythagorean triples.

What are Pythagorean Triples?

Pythagorean triples have been a subject of interest in mathematics for thousands of years. The ancient Greeks were aware of the existence of these triples and used them to construct geometric shapes. A Pythagorean triple is a set of three positive integers, a, b, and c, such that a^2 + b^2 = c^2. This equation is known as the Pythagorean theorem.

Checking the First Set of Side Lengths

The first set of side lengths we will examine is 10, 24, and 26. To determine if this set is a Pythagorean triple, we need to check if the square of the longest side (26) is equal to the sum of the squares of the other two sides (10 and 24).

Step 1: Calculate the square of the longest side (26).
- 26^2 = 676
Step 2: Calculate the sum of the squares of the other two sides (10 and 24).
- 10^2 = 100
- 24^2 = 576
- 100 + 576 = 676

Since the square of the longest side (676) is equal to the sum of the squares of the other two sides (676), the set of side lengths 10, 24, and 26 is a Pythagorean triple.

Checking the Second Set of Side Lengths

The second set of side lengths we will examine is 14, 48, and 49. To determine if this set is a Pythagorean triple, we need to check if the square of the longest side (49) is equal to the sum of the squares of the other two sides (14 and 48).

Step 1: Calculate the square of the longest side (49).
- 49^2 = 2401
Step 2: Calculate the sum of the squares of the other two sides (14 and 48).
- 14^2 = 196
- 48^2 = 2304
- 196 + 2304 = 2500

Since the square of the longest side (2401) is not equal to the sum of the squares of the other two sides (2500), the set of side lengths 14, 48, and 49 is not a Pythagorean triple.

Checking the Third Set of Side Lengths

The third set of side lengths we will examine is 9, 12, and 16. To determine if this set is a Pythagorean triple, we need to check if the square of the longest side (16) is equal to the sum of the squares of the other two sides (9 and 12).

Step 1: Calculate the square of the longest side (16).
- 16^2 = 256
Step 2: Calculate the sum of the squares of the other two sides (9 and 12).
- 9^2 = 81
- 12^2 = 144
- 81 + 144 = 225

Since the square of the longest side (256) is not equal to the sum of the squares of the other two sides (225), the set of side lengths 9, 12, and 16 is not a Pythagorean triple.

Checking the Fourth Set of Side Lengths

The fourth set of side lengths we will examine is 9, 40, and 41. To determine if this set is a Pythagorean triple, we need to check if the square of the longest side (41) is equal to the sum of the squares of the other two sides (9 and 40).

Step 1: Calculate the square of the longest side (41).
- 41^2 = 1681
Step 2: Calculate the sum of the squares of the other two sides (9 and 40).
- 9^2 = 81
- 40^2 = 1600
- 81 + 1600 = 1681

Since the square of the longest side (1681) is equal to the sum of the squares of the other two sides (1681), the set of side lengths 9, 40, and 41 is a Pythagorean triple.

Checking the Fifth Set of Side Lengths

The fifth set of side lengths we will examine is 15, 20, and 25. To determine if this set is a Pythagorean triple, we need to check if the square of the longest side (25) is equal to the sum of the squares of the other two sides (15 and 20).

Step 1: Calculate the square of the longest side (25).
- 25^2 = 625
Step 2: Calculate the sum of the squares of the other two sides (15 and 20).
- 15^2 = 225
- 20^2 = 400
- 225 + 400 = 625

Since the square of the longest side (625) is equal to the sum of the squares of the other two sides (625), the set of side lengths 15, 20, and 25 is a Pythagorean triple.

Q: What is a Pythagorean triple?

A: A Pythagorean triple is a set of three positive integers, a, b, and c, such that a^2 + b^2 = c^2. This equation is known as the Pythagorean theorem.

Q: How are Pythagorean triples used in mathematics?

A: Pythagorean triples are used in various areas of mathematics, including geometry, trigonometry, and algebra. They are also used in real-world applications such as architecture, engineering, and physics.

Q: Can you give an example of a Pythagorean triple?

A: Yes, a classic example of a Pythagorean triple is 3, 4, and 5. This means that if you have a right-angled triangle with sides of length 3 and 4, the length of the hypotenuse (the side opposite the right angle) is 5.

Q: How do you find Pythagorean triples?

A: There are several methods to find Pythagorean triples, including:

Using the Pythagorean theorem: a^2 + b^2 = c^2
Using a formula: a = m^2 - n^2, b = 2mn, c = m^2 + n^2
Using a table or chart: many tables and charts list Pythagorean triples

Q: Are there any patterns or rules for Pythagorean triples?

A: Yes, there are several patterns and rules for Pythagorean triples, including:

The Pythagorean theorem: a^2 + b^2 = c^2
The formula: a = m^2 - n^2, b = 2mn, c = m^2 + n^2
The rule of even and odd numbers: if a and b are both even or both odd, then c is even; if a and b are one even and one odd, then c is odd

Q: Can you give some examples of Pythagorean triples?

A: Yes, here are some examples of Pythagorean triples:

3, 4, 5
5, 12, 13
7, 24, 25
8, 15, 17
9, 40, 41

Q: Are there any real-world applications of Pythagorean triples?

A: Yes, Pythagorean triples have many real-world applications, including:

Architecture: Pythagorean triples are used to design buildings and bridges
Engineering: Pythagorean triples are used to design machines and mechanisms
Physics: Pythagorean triples are used to describe the motion of objects
Computer graphics: Pythagorean triples are used to create 3D models and animations

Q: Can you give some examples of how Pythagorean triples are used in real-world applications?

A: Yes, here are some examples of how Pythagorean triples are used in real-world applications:

In architecture, Pythagorean triples are used to design buildings and bridges. For example, the Great Pyramid of Giza is a right-angled triangle with sides of length 3, 4, and 5.
In engineering, Pythagorean triples are used to design machines and mechanisms. For example, the gears in a bicycle are designed using Pythagorean triples.
In physics, Pythagorean triples are used to describe the motion of objects. For example, the trajectory of a projectile is described using Pythagorean triples.
In computer graphics, Pythagorean triples are used to create 3D models and animations. For example, the characters in a video game are designed using Pythagorean triples.

In conclusion, Pythagorean triples are an important concept in mathematics with many real-world applications. They are used in various areas of mathematics, including geometry, trigonometry, and algebra, and have many practical uses in architecture, engineering, physics, and computer graphics.