Home
Math by Grades Pre-K
Kindergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Grade 6
Grades 7 and 8
Grades 9 and 10
Grades 11 and 12
Math by Topics Arithmetic
Algebra
Geometry
Math Word Problems
Trigonometry
Statistics
Probability
PreCalculus
Calculus
Set Theory
Matrices
Vectors
Math Worksheets Math Worksheets
Interactive Zone
Math in Video Lessons Basic Algebra
Intermediate Algebra
College Algebra
High School Geometry
College Calculus
Linear Algebra
Engineering Math
Singapore Math
Math for Specific Tests SAT Math
ACT Math
GMAT Math
High School, Regents
California Standards
GCSE Maths
A Level Maths
Math Fun and Games Math Trivia
Math Games
Fun Games
Mousehunt Guide
Exam Preparation SAT Preparation
ACT Preparation
GMAT Preparation
Science Biology
Chemistry
Science Projects
High School Biology
High School Chemistry
High School Physics
GCSE Biology
Others English Help
ESL, IELTS, TOEFL
Programming
Animal Facts
Tutoring Services
What's New

   

Pythagorean Triples

In this lesson, we will learn

  • Pythagorean Triples
  • Families of Pythagorean Triples
  • Pythagorean Triples and Right Triangles
  • Solving Problems using the Pythagorean Triples
  • How to generate Pythagorean Triples

 

 

Pythagorean Triples

Pythagorean triples are formed by positive integers a, b and c, such that a2 + b2 = c2. We may write the triple as (a, b, c)

For example, the numbers 3, 4 and 5 form a Pythagorean Triple because 32 + 42 = 52. There are infinitely many Pythagorean triples.

Some examples are


( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)
( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

 

 

Families of Pythagorean Triples

If we multiply each number of a Pythagorean triple by the same number, we form another Pythagorean triple.

For example, (6, 8, 10) is a family of the Pythagorean triple (3, 4, 5) because it can be obtained by 2 × 3 = 6, 2 × 4 = 8, 2 × 5 = 10. If we multiply (3, 4, 5) by 3, we get another triple (9, 12, 15). This can be repeated with different multiples.

Similarly, the family for (5, 12, 13) would be (10, 24, 26), (15, 36, 39) and so on.

 

 

Pythagorean Triples and Right Triangles

When we make a triangle with sides whose lengths are the Pythagorean Triples, the triangle will form a right triangle. This follows from the converse of the Pythagorean Theorem.

5-12-13 triangles

Memorising some common Pythagorean triples can helpful.

For example, if we see that the sides of a triangle form a Pythagorean triple then we know that it is a right triangle.

The following video gives some examples of Pythagorean triples and right triangles.

 

 

Solving problems using the Pythagorean Triples

Pythagorean triples may also help us to find the missing side of a right triangle faster. If two sides of a right triangle form part of a triple then we can know the value of the third side without having to calculate using the Pythagorean theorem.

Example:

Find the value of x

 

Solution:

Check for Pythagorean triple:

Get the ratio of the two given sides:

12 : 20 = 3 : 5 ( divide by 4 )

From the ratio, we know that it is a Pythagorean triple.

So, x = 4 × 4 = 16 cm

 

The following videos show how to solve some GMAT, SAT and ACT questions using the Pythagrorean Triples





 

 

How to generate Pythagrean Trples

The following video will show you how to use an ordinary multiplication table to list infinitely many different examples of Pythagorean Triples

 

 

 

Custom Search

 

We welcome your feedback, comments and questions about this site - please submit your feedback via our Feedback page.

 

© Copyright 2005, 2009 - onlinemathlearning.com
Embedded content, if any, are copyrights of their respective owners.


Useful Links:
More Geometry Help on MathWorld

 

 

 

Custom Search