PYTHAGOREAN TRIPLE

Pythagorean Triple

Definition Of Pythagorean Theorem

A set of three non-zero numbers whose sum of the squares of two numbers is equal to the square of the third number.

More About Pythagorean Theorem

If a, b, c ≠ 0, and a² + b² = c ², then (a, b, c) is called a Pythagorean triple.
Measures of a Pythagorean triple make a right triangle.
If a, b, and c form a Pythagorean triple, then for any positive number n, na, nb, and nc also form a Pythagorean triple i.e. if 3, 4, and 5 form a Pythagorean triple, then 3(3, 4, 5) i.e. 9, 12, and 15 also form a Pythagorean triple.

Video Examples: Pythagorean triples

Example of Pythagorean Theorem

A set (5, 12, 13) is a Pythagorean triple.
5² + 12² = 25 + 144 = 169
13² = 169

Solved Example on Pythagorean Theorem

Ques: Which of the following is a Pythagorean triple?

Choices:

A. 6 m, 8 m, 10 m
B. 5 m, 7 m, 9 m
C. 6 m, 7 m, 8 m
D. 5 m, 6 m, 7 m
Correct Answer: A

Solution:

Step 1: Pythagorean triples are a set of three integers a, b, c, which form the measures of the sides of a right-angled triangle.
Step 2: If the given measures satisfy the condition, c² = a² + b², then the measures make a right triangle.
Step 3: Here, the measures 6 m, 8 m, and 10 m satisfy the condition. (10² = 6² + 8 ² ⇒ 100 = 36 + 64 = 100)
Step 4: Thus, the measures 6 m, 8 m, and 10 m make a right triangle and thus it is a Pythagorean triple.

Quick Summary

Pythagorean triples represent the side lengths of right triangles.
If (a, b, c) is a Pythagorean triple, then (na, nb, nc) is also a Pythagorean triple for any positive integer n.
Common Pythagorean triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

\[ a^2 + b^2 = c^2 \]

🍎 Teacher Insights

Emphasize the connection between Pythagorean triples and the Pythagorean Theorem. Use visual aids, such as right triangles drawn to scale, to illustrate the concept. Encourage students to find and verify their own Pythagorean triples.

🎓 Prerequisites

Basic Algebra
Understanding of Squares and Square Roots
Pythagorean Theorem

Check Your Knowledge

Q1: Which of the following is a Pythagorean triple?

A. (1, 2, 3) B. (2, 3, 4) C. (3, 4, 5) D. (4, 5, 6)

Q2: If (6, 8, x) is a Pythagorean triple, what is x?

A. 9 B. 10 C. 11 D. 12

Frequently Asked Questions

Q: What is the smallest Pythagorean triple?
A: The smallest Pythagorean triple is (3, 4, 5).

Q: How can I generate Pythagorean triples?
A: There are formulas to generate Pythagorean triples. One common method involves using two integers m and n, where m > n > 0, and calculating a = m^2 - n^2, b = 2mn, and c = m^2 + n^2.