Question

Determine if the following lengths are Pythagorean triples: 9, 40 and 41.

Answer 1

Hint:
Here we need to check whether the given triplets are Pythagorean triples or not. For that, we will use the basic definition of Pythagoras theorem. We will find the greatest number among the given numbers. Then we will find the sum of squares of two lengths and check if it is equal to the square of greatest number or not. If it satisfies the condition the numbers are Pythagorean triples otherwise not.

Complete step by step solution:
The given triplets are 9, 40 and 41.
Now, we will use the basic definition of Pythagoras theorem which states that the square of the longest side of the right angled triangle is equal to the sum of the squares of the other two sides of the right angled triangle.
We can see that the 41 is of greatest length among the triplets.
So, we can write the mathematical equation using the Pythagoras theorem as:
\[{41^2} = {40^2} + {9^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow 1681 = 1600 + 81\]
On adding the numbers, we get
\[ \Rightarrow 1681 = 1681\]

As both sides of the equation are equal, so we can say that this is satisfying the Pythagoras theorem. So, the given triplets are Pythagorean triples.

Note:
Three numbers can be called as Pythagorean triplets only if they satisfy the Pythagoras theorem. Pythagoras theorem is used to find the sides of a right angled triangle. This theorem is only applicable to the right angled triangle only. Here, we need to keep in mind that the square of the largest number must be equal to the sum of the square of the other two numbers. If we swap any of the numbers then the theorem will not be satisfied.