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List all the perfect squares between $ 1 $ and $ 50 $ . Which of these numbers form a Pythagorean triplet? How many triplets can you find?

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Last updated date: 19th Jul 2024
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Answer
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Hint: As we know that a perfect square is a number that can be expressed as the square of a number from the number system. For example, $ 25 $ is a perfect square. It is the square of the natural number $ 5 $ i.e. $ {5^2} = 25 $ . Now by applying this we will write all the perfect squares between $ 1 $ and $ 50 $ .

Complete step-by-step answer:
We have to find the perfect squares between $ 1 $ and $ 50 $ .
By applying the above definition we have the perfect squares $ 1,4,9,16,25,36,49 $ .
So there are total $ 7 $ perfect squares between $ 1 $ and $ 50 $ .
Now we know that a Pythagorean triplet consists of three positive terms $ a,b $ and $ c $ , such that $ {a^2} + {b^2} = {c^2} $ .
We can write these perfect squares $ 1,4,9,16,25,36,49 $ as $ {1^2},{2^2},{3^2},{4^2},{5^2},{6^2},{7^2} $ .
From the above we have to choose three numbers such that $ {a^2} + {b^2} = {c^2} $ .
We can see that there is only one such pair that forms the Pythagorean triplet which is $ 3,4,5 $.
It can be written as $ {3^2} + {4^2} = {5^2} $ . On further solving by squaring we have $ 9 + 16 = 25 \Rightarrow 25 = 25 $ .
Hence the required Pythagorean triplet is $ (3,4,5) $ .
So, the correct answer is “ $ (3,4,5) $ ”.

Note: We should note that $ (3,4,5) $ is the most known and smallest Pythagorean triplet. We should know that if a triangle has one angle which is right angle i.e. $ {90^ \circ } $ , then there is the Pythagoras theorem. Let us take $ r $ to be the hypotenuse and the other two sides are $ p $ and $ q $ , then it states that $ {p^2} + {q^2} = {r^2} $ . We can define it as the sum of the squares of the other two sides is the same as the square of the longest side.