Question

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

Answer 1

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.