Question

Which of the following is a Pythagorean triplet?
(a) (2, 3, 5)
(b) (5, 7, 9)
(c) (6, 9, 11)
(d) (8, 15, 17)

Answer 1

Hint:Here, we will use the concept of Pythagorean triplet and check each of the given options. A Pythagorean triplet refers to a triplet that is a set of three numbers that satisfy the Pythagoras theorem, i.e. the sum of squares of two of the numbers is equal to the square of the third number.

Complete step-by-step answer:
The term Pythagorean triplet has been derived from the Pythagoras theorem stating that every right angled triangle, i.e. a triangle in which one of the angle measures 90 degrees has side lengths satisfying the formula:
${{\left( hypotenuse \right)}^{2}}={{\left( base \right)}^{2}}+{{\left( perpendicular \right)}^{2}}$
A Pythagorean triplet consists of three positive integers a, b and c such that ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$. A triangle whose sides form a Pythagorean triplet is necessarily a right angled triangle.
We know that the hypotenuse is the longest side in a right angled triangle. So, while checking for the given options, we have to see whether the square of the greatest number is equal to the sum of the squares of the other two numbers.
The first option is (2, 3, 5):
We have, ${{2}^{2}}=4$ , ${{3}^{2}}=9$ and ${{5}^{2}}=25$
Now, 4+9 = 13, which is not equal to 25.
So, ${{5}^{2}}\ne {{2}^{2}}+{{3}^{2}}$
Therefore, these numbers do not form a Pythagorean triplet.
The second option is (5, 7, 9):
We have, ${{5}^{2}}=25$, ${{7}^{2}}=49$ and ${{9}^{2}}=81$
Now, 25+ 49=74, which is not equal to 81.
So, ${{9}^{2}}\ne {{5}^{2}}+{{7}^{2}}$
Therefore, these numbers do not form a Pythagorean triplet.
The third option is (6, 9, 11):
We have ${{6}^{2}}=36$, ${{9}^{2}}=81$ and ${{11}^{2}}=121$
Now, 36+81=127, which is not equal to 121.
So, ${{11}^{2}}\ne {{6}^{2}}+{{9}^{2}}$
Therefore, these numbers do not form a Pythagorean triplet.
The fourth option is (8, 15, 17):
We have, ${{8}^{2}}=64$, ${{15}^{2}}=225$ and ${{17}^{2}}=289$
Now, 64+225=289 which is equal to the square of 17.
So, ${{17}^{2}}={{8}^{2}}+{{15}^{2}}$
Therefore, these numbers form a Pythagorean triplet.
Hence, option (d) is the correct answer.

Note:Students should note that while checking for Pythagorean triplets, we always check whether the square of the greatest number (which represents the hypotenuse of the right angled triangle) is equal to the sum of squares of the other two numbers.