What is a Pythagorean Triplet?

S3-SA4-0324

Grade Level:

Class 6

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

Definition

What is it?

A Pythagorean Triplet is a set of three positive whole numbers (integers) that perfectly satisfy 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. If a set of three numbers (a, b, c) makes a^2 + b^2 = c^2 true, they form a Pythagorean Triplet.

Simple Example

Quick Example

Imagine you have three friends who scored marks in their Maths test: Friend A got 3 marks, Friend B got 4 marks, and Friend C got 5 marks. If we check if these numbers form a Pythagorean Triplet: 3^2 + 4^2 = 9 + 16 = 25. And 5^2 = 25. Since 25 = 25, (3, 4, 5) is a Pythagorean Triplet!

Worked Example

Step-by-Step

Let's check if the numbers (6, 8, 10) form a Pythagorean Triplet.
1. Identify the largest number. Here, it is 10. This will be 'c'. The other two numbers, 6 and 8, will be 'a' and 'b'.
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2. Calculate the square of the first smaller number (a^2). So, 6^2 = 6 * 6 = 36.
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3. Calculate the square of the second smaller number (b^2). So, 8^2 = 8 * 8 = 64.
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4. Add the squares of the two smaller numbers (a^2 + b^2). So, 36 + 64 = 100.
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5. Calculate the square of the largest number (c^2). So, 10^2 = 10 * 10 = 100.
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6. Compare the sum from step 4 with the square from step 5. Is a^2 + b^2 = c^2? Yes, 100 = 100.
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Answer: Since 6^2 + 8^2 = 10^2, the numbers (6, 8, 10) form a Pythagorean Triplet.

Why It Matters

Pythagorean Triplets are super useful in many fields! Engineers use them to design strong buildings and bridges, ensuring stability. Computer scientists use them in graphics and game development to calculate distances and positions. Even in fields like Data Science, understanding these relationships helps in analyzing spatial data, making them foundational for many cool careers.

Common Mistakes

MISTAKE: Not identifying the largest number as 'c' (the hypotenuse). Students sometimes pick any number for 'c'. | CORRECTION: Always make sure the largest number in the triplet is taken as 'c' because it represents the hypotenuse, which is always the longest side in a right-angled triangle.

MISTAKE: Forgetting to square the numbers. Students might just add the numbers instead of their squares. | CORRECTION: Remember the formula is a^2 + b^2 = c^2, which means you must multiply each number by itself before adding or comparing.

MISTAKE: Thinking any three numbers form a triplet. Students might assume if numbers are close, they will form a triplet. | CORRECTION: Only specific sets of three whole numbers satisfy the condition a^2 + b^2 = c^2 exactly. If the sum of the squares of the two smaller numbers is not exactly equal to the square of the largest number, it's not a Pythagorean Triplet.

Practice Questions

Try It Yourself

QUESTION: Is (5, 12, 13) a Pythagorean Triplet? | ANSWER: Yes, because 5^2 + 12^2 = 25 + 144 = 169, and 13^2 = 169. So, 169 = 169.

QUESTION: Which of these is NOT a Pythagorean Triplet: (7, 24, 25) or (8, 15, 17) or (4, 5, 6)? | ANSWER: (4, 5, 6) is NOT a Pythagorean Triplet. Because 4^2 + 5^2 = 16 + 25 = 41, but 6^2 = 36. Since 41 is not equal to 36.

QUESTION: A carpenter needs to check if a wooden frame is perfectly right-angled. The sides measure 9 cm, 12 cm, and 15 cm. Is the frame perfectly right-angled? | ANSWER: Yes, the frame is perfectly right-angled. Because 9^2 + 12^2 = 81 + 144 = 225, and 15^2 = 225. Since 225 = 225, the sides form a Pythagorean Triplet, indicating a right angle.

MCQ

Quick Quiz

Which set of numbers forms a Pythagorean Triplet?

(2, 3, 4)

(6, 7, 8)

(9, 12, 15)

(1, 2, 3)

The Correct Answer Is:

For (9, 12, 15): 9^2 + 12^2 = 81 + 144 = 225, and 15^2 = 225. Since 225 = 225, it is a Pythagorean Triplet. The other options do not satisfy the a^2 + b^2 = c^2 condition.

Real World Connection

In the Real World

When a civil engineer in India designs a new flyover or a building, they often use Pythagorean Triplets to ensure that corners are perfectly square (90 degrees). For example, when laying out foundations, they measure distances to form a (3, 4, 5) or (6, 8, 10) triangle to guarantee the right angles, ensuring the structure is strong and safe. This is also used by architects to draw accurate blueprints for your home or school.

Key Vocabulary

Key Terms

PYTHAGOREAN THEOREM: A rule stating a^2 + b^2 = c^2 for right-angled triangles | RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle | WHOLE NUMBERS: Counting numbers like 1, 2, 3, and so on (no fractions or decimals)

What's Next

What to Learn Next

Great job understanding Pythagorean Triplets! Now you're ready to explore the Pythagorean Theorem in more detail, learning how to find a missing side of a right-angled triangle when you know the other two. This will help you solve even more exciting real-world problems!