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Last updated date: 02nd Dec 2023
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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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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 $50$ .

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.