Pythagorean triples are the three positive integers that completely satisfy the Pythagorean theorem. The theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs of the right triangle. These three sides of the right triangle form the Pythagorean triples. Let us learn to generate a few Pythagorean triples quickly.
|2.||Examples of Pythagorean Triples|
|3.||Pythagorean Triples Proof|
|4.||List of Pythagorean Triples|
|5.||Solved Examples on Pythagorean Triples|
|6.||Practice Questions on Pythagorean Triples|
|7.||FAQs on Pythagorean Triples|
The general formula for Pythagorean triples can be shown as, a2 + b2 = c2, where a, b, and c are the positive integers that satisfy this equation, where 'c' is the "hypotenuse" or the longest side of the triangle and a and b are the other two legs of the right-angled triangle. The Pythagorean triples are represented as (a,b, c). The most popular example of Pythagorean triples is (3, 4, 5). We can verify that 3, 4 and 5 satisfy the equation a2 + b2 = c2. Let's see how!
32 + 42 = 52 ⇒ 9 + 16 = 25
Examples of Pythagorean Triples
The set of the Pythagorean triples is endless. The first known Pythagorean triples is (3, 4, and 5). We can generate a few more triples by scaling them up in the following manner. We can create as many triples as possible by taking values for n.
|n||(3n, 4n, 5n)|
|2||(6, 8, 10)|
|3||(9, 12, 15)|
|4||(12, 16, 20)|
Pythagorean Triples Proof
Let us consider Pythagorean triples (9, 40, 41) for which we can verify the Pythagorean formula (Hypotentuse2 = side12 + side22). The hypotenuse of the right-angled triangle is the longest side = 41
Hypotentuse2 = 412 = 1681
The other two sides of the right angled triangle = 9 and 40
side12 + side22 = 92 + 402 = 81 + 1600 = 1681
Thus for any 3 Pythagorean Triples, we can verify the Pythagorean formula.
Tips and Tricks
- If the number (n) is odd, the Pythagorean triples is of the form, (n, (n2/2 - 0.5) and (n2/2 + 0.5)). For example, consider 5. The triples are (5, 25/2 - 0.5, 25/2 + 0.5) Finally, we get (5, 12 and 13)
- If the number (n) is even, Pythagorean triples is of the form = n, (n/2)2-1), ((n/2)2+1). For example, consider 6. The triples are (6, (3)2 - 1, (3)2 + 1) Finally, we get (6, 8, and 10)
List of Pythagorean Triples
Given here is a list of a few Pythagorean Triples.
|(3, 4, 5)||(5,12,13)||(7, 24, 25)|
|(8, 15, 17)||(9, 40, 41)||(11, 60, 61)|
|(12,35, 37)||(13, 84, 85)||(15, 112, 113)|
|(16, 63, 65)||(17,144, 145)||(19, 180, 181)|
|(20, 21, 29)||(20, 99 ,101)||(21, 220,221)|
How to Generate Pythagorean Triples?
You can generate a Pythagorean triple by employing a formula, which is also known as the Pythagorean triple checker. Let us assume any 2 integers m and n, which will help us in generating the Pythagorean formula. Now, the lengths of our sides will be a, b, and c. We will use m and n in order to find the exact values of the sides.
- The length of side a is determined by defining the difference between the squares of m and n that we can express as an equation, as a = m2 - n2
- By doubling the m and n as a product, the length of side b is determined. We have b = 2mn in the equation form.
- Finally, by having the sum of the squares of m and n, the length of side c is computed. This can be written simply in the equation as c = m2 + n2
Let us try generating a Pythagorean triple using the two integers 2 and 3. Since m>n, m = 3, and n = 2. Since we know the values of m and n, it's time to substitute those values into the formulas of a, b, and c, to get the sides of the right triangle.
Computing a: a = m2 - n2
a = 32 - 22
9 - 4 = 5
a = 5
Computing b: b = 2mn
b = 2 × 3 × 2
b = 12
Computing c: c = m2 + n2
c = 32 + 22
c = 13
Let’s see if our values for a =5, b=12, and c=13 satisfy the Pythagorean triple equation, which is a2 + b2 = c2
LHS: 52 + 122 = 25 +144 = 169
RHS : 132 = 169
Yes, it does! Therefore, (5, 12, 13) are Pythagorean triples.
- Any three positive integers which satisfy the formula of a2 + b2 = c2 are known as Pythagorean triples.
- If any number of a Pythagorean triple is given, then the other two numbers can be generated by using, a = m2 - n2, b = 2mn, and c = m2 + n2.
- Pythagorean triples can not be in decimals.
- If (a, b, c) is a Pythagorean triplet, then ta, tb, tc will also form a Pythagorean triplet; where t is any positive integer.
Related Articles on Pythagorean Triples
Check out the articles below to know more about terminologies related to Pythagorean Triples.
Solved Examples on Pythagorean Triples
Example 1: John's mother asked him to place the ladder at such a distance from the wall, such that the 13 feet tall ladder's head falls exactly on the top of the 12 feet wall. Can you help John to find the distance of the ladder from the wall?
Solution: Let the required distance be 'x' feet. Here, the ladder, the wall, and the ground form a right-angled triangle. The ladder is the hypotenuse of the triangle. x2 +122 = 132
x2 = 132 -122
169-144 = 5 feet
Therefore, the distance of the ladder from the wall is 5 feet.
Example 2: Joey tried a new route to reach his school today. He walked 6 blocks to the north, and then 8 blocks to the west. Can you find how far is his school from his home?
Solution: The distance from the school to home is the length of the hypotenuse. Let c be the missing distance from school to home and a = 6, b = 8.
c2 = a2 + b2
c2 = 62 + 82
36 + 64 = 100
c = 10
Therefore, the distance from school to home is 10 blocks.
Example 3: Check if 7, 12 and 19 are Pythagorean triples. Also find, what could be the Pythagorean triples of 7?
Solution: To be the Pythagorean triples, we check if they satisfy the equation c2 = a2 + b2
192 = 361
72 + 122 = 49 + 144
Thus we find (7, 12, 19) don't satisfy the equation. They are not Pythagorean Triples. To generate Pythagorean Triples, we check if the number is odd or even. Here 7 is odd. If n is odd, the other two triples must be (n, (n2/2 - 0.5) and (n2/2 + 0.5)).
(n2/2 - 0.5) = 49/2 - 0.5 = 24
(n2/2 + 0.5) = 49/2 + 0.5 = 25
Therefore, (7, 24, 25) are the Pythagorean triples.
FAQs on Pythagorean Triples
What are Pythagorean Triples?
The Pythagorean triples are the 3 positive integers that fit in the rule a2 = b2 + c2.
How to Find Pythagorean Triples?
If a number is odd, then find if the triples are (n, (n2/2 - 0.5) and (n2/2 + 0.5)). If the number is even, then find if the triples are (n, (n/2 - 1) and (n/2 +1)).
What are the Pythagorean Triples of 6?
The Pythagorean triples for 6 are (6, 8, 10) as they satisfy the equation a2 = b2 + c2.
Can Pythagorean Triples have Decimals?
Pythagorean triples are the positive integers that fit the formula of the Pythagorean theorem. These are natural numbers that cannot be in decimals.
What are the 5 Most Common Pythagorean Triples?
The 5 most common Pythagorean triples are (3, 4, 5), (5, 12, 13), (6, 8, 10), (9, 12, 15), and (15, 20, 25).
How do you Find the Pythagorean Triples of 12?
Here, 12 is even. So the triples are (12, (12/2)2 -1, (12/2)2 +1). Therefore, we have the Pythagorean triples of 12 as (12, 35, 37).
How do you Find the Pythagorean Triples of 7?
Here, 7 is odd, So the triples are (7, (72 / 2 -0.5) , (72 / 2 -0.5)). Therefore, we have the Pythagorean Triples of 7 as (7, 24, 25).