Let us consider the right triangle shown below.
We can use the following formulas to generate a Pythagorean triple.
Hypotenuse = m² + n²
Leg1 = m² - n²
Leg2 = 2mn
In the above formulas, always m has to be greater than n.
That is,
m > n
In the above formulas, we can take any positive values for m and n to get the three sides of a right triangle such that m > n.
For example, let m = 3 and n = 2.
Then, we have
Hypotenuse = m2 + n2 = 32 + 22 = 9 + 4 = 13
Leg1 = m2 - n2 = 32 - 22 = 9 - 4 = 5
Leg2 = 2mn = 2(3)(2) = 12
Now, we can check whether 13, 5, 12 be the three sides of the right angle triangle using Pythagorean theorem.
That is,
(Hypotenuse)2 = (Leg1)2 + (leg2)2
132 = 52 + 122
169 = 25 + 144
169 = 169
The three values 13, 5 and 12 satisfy the Pythagorean Theorem.
Therefore, the values 13, 5 and 12 form a Pythagorean triple.
Example 1 :
Using 5 and 6, create the lengths of three sides of a right triangle.
Solution :
Since m > n, we can take m = 6 and n = 5.
Then, we have
Hypotenuse = m2 + n2 = 62 + 52 = 36 + 25 = 61
Leg1 = m2 - n2 = 62 - 52 = 36 - 25 = 11
Leg2 = 2mn = 2(6)(5) = 60
So, the lengths of three sides of the right triangle are
61, 11 and 60
Example 2 :
Using 4 and 3, create the lengths of three sides of a right triangle.
Solution :
Since m > n, we can take m = 4 and n = 3.
Then, we have
Hypotenuse = m2 + n2 = 42 + 32 = 16 + 9 = 25
Leg1 = m2 - n2 = 42 - 32 = 16 - 9 = 7
Leg2 = 2mn = 2(4)(3) = 24
So, the lengths of three sides of the right triangle are
25, 7 and 24
Example 3 :
Using 1 and 2, create the lengths of three sides of a right triangle.
Solution :
Since m > n, we can take m = 2 and n = 1.
Then, we have
Hypotenuse = m2 + n2 = 22 + 12 = 4 + 1 = 5
Leg1 = m2 - n2 = 22 - 12 = 4 - 1 = 3
Leg2 = 2mn = 2(2)(1) = 4
So, the lengths of three sides of the right triangle are
5, 3 and 4
Example 4 :
Using 6 and 7, create the lengths of three sides of a right triangle.
Solution :
Since m > n, we can take m = 7 and n = 6.
Then, we have
Hypotenuse = m2 + n2 = 72 + 62 = 49 + 36 = 85
Leg1 = m2 - n2 = 72 - 62 = 49 - 36 = 13
Leg2 = 2mn = 2(7)(6) = 84
So, the lengths of three sides of the right triangle are
85, 13 and 84
Example 5 :
Using 11 and 6, create the lengths of three sides of a right triangle.
Solution :
Since m > n, we can take m = 11 and n = 6.
Then, we have
Hypotenuse = m2 + n2 = 112 + 62 = 121 + 36 = 157
Leg1 = m2 - n2 = 112 - 62 = 121 - 36 = 85
Leg2 = 2mn = 2(11)(6) = 132
So, the lengths of three sides of the right triangle are
157, 85 and 132
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