What is Pythagorean Triplets?

S3-SA4-0111

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Pythagorean Triplets are a set of three positive whole numbers (integers) that perfectly fit the Pythagorean theorem. This theorem states that in a right-angled triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides (legs). If three numbers 'a', 'b', and 'c' satisfy a^2 + b^2 = c^2, they form a Pythagorean Triplet.

Simple Example

Quick Example

Imagine you have three friends who scored marks in a test: 3, 4, and 5. If we check if these marks form a Pythagorean Triplet, we see if 3^2 + 4^2 equals 5^2. Since 9 + 16 = 25, and 5^2 is also 25, these numbers (3, 4, 5) form a Pythagorean Triplet! This means they could be the side lengths of a right-angled triangle.

Worked Example

Step-by-Step

Let's check if the numbers 5, 12, and 13 form a Pythagorean Triplet.
---Step 1: Identify the two smaller numbers and the largest number. Here, 5 and 12 are the smaller numbers, and 13 is the largest.
---Step 2: Square the first smaller number. 5^2 = 5 * 5 = 25.
---Step 3: Square the second smaller number. 12^2 = 12 * 12 = 144.
---Step 4: Add the squares of the two smaller numbers. 25 + 144 = 169.
---Step 5: Square the largest number. 13^2 = 13 * 13 = 169.
---Step 6: Compare the sum from Step 4 with the square from Step 5. Since 169 = 169, they are equal.
---Answer: Yes, the numbers 5, 12, and 13 form a Pythagorean Triplet because 5^2 + 12^2 = 13^2.

Why It Matters

Pythagorean Triplets are fundamental in many fields like engineering, architecture, and computer graphics. Architects use them to ensure buildings have perfectly square corners, while game developers use them to calculate distances and positions of objects. Understanding them helps build a strong base for future studies in AI/ML and data science, where spatial relationships are crucial.

Common Mistakes

MISTAKE: Students often forget to square the numbers before adding them, simply adding the numbers directly. | CORRECTION: Always remember to square (multiply by itself) each of the two smaller numbers first, then add their results. Finally, square the largest number.

MISTAKE: Assuming any three numbers form a triplet without checking the condition a^2 + b^2 = c^2. | CORRECTION: You must always test the condition by squaring the two smaller numbers and adding them, then squaring the largest number and comparing the results. They must be exactly equal.

MISTAKE: Incorrectly identifying the hypotenuse (the 'c' in a^2 + b^2 = c^2) when given three numbers. | CORRECTION: The largest number among the three given numbers will always be the hypotenuse. The other two smaller numbers are the legs.

Practice Questions

Try It Yourself

QUESTION: Do the numbers 6, 8, and 10 form a Pythagorean Triplet? | ANSWER: Yes

QUESTION: If two sides of a right-angled triangle are 7 cm and 24 cm, what would be the length of the hypotenuse for it to form a Pythagorean Triplet? | ANSWER: 25 cm

QUESTION: Check if (10, 24, 26) is a Pythagorean Triplet. If yes, find another triplet by multiplying each number by 2. | ANSWER: Yes, (10, 24, 26) is a triplet. The new triplet is (20, 48, 52).

MCQ

Quick Quiz

Which of the following sets of numbers is a Pythagorean Triplet?

1, 2, 2003

4, 5, 2006

8, 15, 17

2, 3, 2004

The Correct Answer Is:

For option C, 8^2 + 15^2 = 64 + 225 = 289. And 17^2 = 289. Since 289 = 289, it is a Pythagorean Triplet. The other options do not satisfy the condition a^2 + b^2 = c^2.

Real World Connection

In the Real World

When a civil engineer in India designs a bridge or a building, they often use Pythagorean Triplets to ensure that the corners are perfectly square (90 degrees). For example, they might measure 3 units along one wall and 4 units along the other. If the diagonal distance between those two points is exactly 5 units, they know the corner is square. This is crucial for safety and stability in construction, from a small house to a large metro station.

Key Vocabulary

Key Terms

Pythagorean Theorem: A rule in geometry that states a^2 + b^2 = c^2 for a right-angled triangle | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle | Legs: The two shorter sides of a right-angled triangle that form the right angle | Square: The result of multiplying a number by itself (e.g., 4 squared is 4 * 4 = 16) | Integer: A whole number (positive, negative, or zero)

What's Next

What to Learn Next

Great job learning about Pythagorean Triplets! Next, you can explore 'Applications of Pythagorean Theorem' to see how these triplets are used to solve real-world problems involving heights, distances, and navigation. This will deepen your understanding and show you the practical power of this concept.