What are Pythagorean Triplets?

S3-SA4-0288

Grade Level:

Class 6

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

Definition

What is it?

Pythagorean Triplets are a special set of three positive whole numbers (a, b, c) that perfectly fit the Pythagorean theorem: a^2 + b^2 = c^2. In simple words, if you square the first two numbers and add them, you get the square of the third number. These triplets are super useful for understanding right-angled triangles.

Simple Example

Quick Example

Imagine you have three friends who scored marks in a math test: Friend A got 3 marks, Friend B got 4 marks, and Friend C got 5 marks. If you square Friend A's marks (3^2 = 9) and Friend B's marks (4^2 = 16) and add them (9 + 16 = 25), you get the square of Friend C's marks (5^2 = 25). So, (3, 4, 5) is a Pythagorean Triplet!

Worked Example

Step-by-Step

Let's check if (8, 15, 17) is a Pythagorean Triplet. We need to see if 8^2 + 15^2 = 17^2.
---Step 1: Square the first number (a). 8^2 = 8 * 8 = 64.
---Step 2: Square the second number (b). 15^2 = 15 * 15 = 225.
---Step 3: Add the squares of the first two numbers. 64 + 225 = 289.
---Step 4: Square the third number (c). 17^2 = 17 * 17 = 289.
---Step 5: Compare the results. Since 289 = 289, the equation holds true.
---Answer: Yes, (8, 15, 17) is a Pythagorean Triplet.

Why It Matters

Pythagorean Triplets are fundamental in many fields, from building houses to designing computer games. Engineers use them to ensure structures are stable, and computer scientists use them in graphics and navigation systems. Understanding these triplets helps you think like a problem-solver in careers like architecture, game development, or even space science at ISRO!

Common Mistakes

MISTAKE: Not squaring the numbers before adding them. Students sometimes add a + b = c instead of a^2 + b^2 = c^2. | CORRECTION: Always remember to square each number first, then add the squares of the two smaller numbers and compare to the square of the largest number.

MISTAKE: Assuming any three numbers that add up to a certain value are a triplet. For example, thinking (2, 3, 5) is a triplet because 2+3=5. | CORRECTION: Pythagorean Triplets are based on the sum of SQUARES, not just the numbers themselves. Always check a^2 + b^2 = c^2.

MISTAKE: Mixing up which number is 'c'. Students might put the smallest number as 'c'. | CORRECTION: In a Pythagorean Triplet (a, b, c), 'c' is always the largest number. It's the hypotenuse in a right-angled triangle, which is always the longest side.

Practice Questions

Try It Yourself

QUESTION: Is (6, 8, 10) a Pythagorean Triplet? | ANSWER: Yes

QUESTION: Check if (7, 24, 25) is a Pythagorean Triplet. Show your steps. | ANSWER: 7^2 = 49, 24^2 = 576. 49 + 576 = 625. 25^2 = 625. Since 625 = 625, yes, it is a Pythagorean Triplet.

QUESTION: If (5, 12, x) is a Pythagorean Triplet, what is the value of x? | ANSWER: x = 13 (Because 5^2 + 12^2 = 25 + 144 = 169, and sqrt(169) = 13)

MCQ

Quick Quiz

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

(1, 2, 3)

(5, 12, 13)

(4, 5, 6)

(2, 3, 4)

The Correct Answer Is:

For (5, 12, 13): 5^2 + 12^2 = 25 + 144 = 169. Also, 13^2 = 169. Since 169 = 169, it is a Pythagorean Triplet. The other options do not satisfy a^2 + b^2 = c^2.

Real World Connection

In the Real World

Imagine masons building a wall or framing a door in your home. They often use the (3, 4, 5) rule to make sure the corners are perfectly square (90 degrees). By measuring 3 units along one side, 4 units along the other, and checking if the diagonal is 5 units, they can guarantee a right angle without a fancy tool. This simple concept ensures buildings are strong and straight!

Key Vocabulary

Key Terms

Pythagorean Theorem: a fundamental rule relating the sides of a right-angled triangle | Square: multiplying a number by itself (e.g., 3^2 = 9) | Whole Numbers: positive counting numbers like 1, 2, 3, ... | Right-angled Triangle: a triangle with one angle exactly 90 degrees | Hypotenuse: the longest side of a right-angled triangle, opposite the 90-degree angle

What's Next

What to Learn Next

Great job understanding Pythagorean Triplets! Now you're ready to explore the full Pythagorean Theorem and how it applies to right-angled triangles. This will help you find missing side lengths in triangles and solve more complex geometry problems.