What is a Pythagorean Triplet? | Simple Explanation for Class 6

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What is the Pythagorean Triplet?

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 (like 3, 4, 5) that fit the Pythagorean theorem. This theorem says 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. So, for numbers a, b, and c, if 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 math test: one got 3 marks, another got 4 marks, and the third got 5 marks. If we check if 3^2 + 4^2 = 5^2, we get 9 + 16 = 25. Since 25 = 25, these marks (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 largest number. Here, the largest number is 13.

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Step 2: Square the two smaller numbers. The smaller numbers are 5 and 12.
5^2 = 5 * 5 = 25
12^2 = 12 * 12 = 144

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Step 3: Add the squares of the two smaller numbers.
25 + 144 = 169

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Step 4: Square the largest number.
13^2 = 13 * 13 = 169

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Step 5: Compare the results from Step 3 and Step 4.
Is 169 equal to 169? Yes, it is!

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Answer: Since 5^2 + 12^2 = 13^2 (25 + 144 = 169), the numbers 5, 12, and 13 form a Pythagorean Triplet.

Why It Matters

Pythagorean Triplets are super useful in many fields! Engineers use them to build stable structures like bridges and buildings, ensuring they don't fall. In computer graphics, they help create realistic 3D shapes. Even in cryptography, which keeps your online messages secure, these concepts are used to design strong codes.

Common Mistakes

MISTAKE: Not squaring the numbers before adding them, e.g., checking if a + b = c. | CORRECTION: Always remember to square each number (multiply it by itself) before adding the two smaller ones and comparing with the square of the largest.

MISTAKE: Assuming any three numbers form a triplet without checking the condition a^2 + b^2 = c^2. | CORRECTION: You must always verify the Pythagorean theorem. Not all sets of three numbers will satisfy the condition.

MISTAKE: Incorrectly identifying the largest number (hypotenuse) to be 'c'. | CORRECTION: Always make sure 'c' is the largest number in the triplet, as it represents the hypotenuse in a right-angled triangle.

Practice Questions

Try It Yourself

QUESTION: Do the numbers 6, 8, 10 form a Pythagorean Triplet? | ANSWER: Yes, because 6^2 + 8^2 = 36 + 64 = 100, and 10^2 = 100. Since 100 = 100, they form a triplet.

QUESTION: Check if 7, 24, 25 form a Pythagorean Triplet. | ANSWER: Yes, because 7^2 + 24^2 = 49 + 576 = 625, and 25^2 = 625. Since 625 = 625, they form a triplet.

QUESTION: If the two smaller sides of a right-angled triangle are 9 cm and 12 cm, what would be the length of the longest side if it were part of a Pythagorean Triplet? | ANSWER: Let the longest side be x. Then 9^2 + 12^2 = x^2. 81 + 144 = x^2. 225 = x^2. So, x = sqrt(225) = 15 cm. The triplet would be 9, 12, 15.

MCQ

Quick Quiz

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

2003-01-02T00:00:00.000Z

2006-04-05T00:00:00.000Z

2017-08-15T00:00:00.000Z

2004-02-03T00:00:00.000Z

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 new flyover or a building, they often use Pythagorean Triplets to make sure the angles are perfectly 90 degrees and the structure is strong. For example, they might use a 3-4-5 rule with measuring tapes to set out square corners on a construction site, just like the masons who build your house use similar tricks to ensure straight walls.

Key Vocabulary

Key Terms

Pythagorean Theorem: A rule stating that in a right-angled triangle, a^2 + b^2 = c^2 | Right-angled triangle: A triangle with one angle exactly 90 degrees | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle | Square: Multiplying a number by itself (e.g., 4 squared is 4x4=16) | Triplet: A set of three numbers.

What's Next

What to Learn Next

Great job understanding Pythagorean Triplets! Next, you can explore the Pythagorean Theorem in more detail and learn how to use it to find missing side lengths in right-angled triangles. This will help you solve many practical problems in geometry and beyond.