Question

Write statement of Pythagoras theorem and show that 6, 8 and 10 are Pythagorean triples.

Answer 1

Hint: First, write the definition of Pythagoras theorem. For the second part, identify the largest number and square it. Then find the sum of the squares of the other two numbers. Equate the above two resultants to arrive at the final answer.

Complete step-by-step answer:
In this question, we need to write the statement of Pythagoras theorem and show that 6, 8 and 10 are Pythagorean triples.
Let us first write the statement of Pythagoras theorem.
Pythagoras Theorem states that the in a right angled triangle Square of the length of the hypotenuse of the right angled triangle is equal to the sum of the squares of the lengths of other two sides of the right angled triangle.
i.e. ${{a}^{2}}={{b}^{2}}+{{c}^{2}}$, where a is the hypotenuse and the longest side of the right angled triangle and b and c are the other two sides of the right angled triangle.
A Pythagorean triplet consists of three positive integers a, b, and c, which satisfy the above condition ${{a}^{2}}={{b}^{2}}+{{c}^{2}}$.
Let us now show that 6, 8 and 10 are Pythagorean triples.
We know that the longest side of a right angled triangle is its hypotenuse.
So, in the given triplet, 10 is the largest number and hence, it will be the hypotenuse.
So, hypotenuse = 10
On squaring both sides, we will get the following:
\[{{\left( hypotenuse \right)}^{2}}={{10}^{2}}=100\] …(1)
Now, we will look into the other sides:
\[{{6}^{2}}+{{8}^{2}}=36+64=100\] …(2)
Thus, from equations (1) and (2), we will get the following:
${{10}^{2}}={{6}^{2}}+{{8}^{2}}$
This satisfies the condition for Pythagorean triplets ${{a}^{2}}={{b}^{2}}+{{c}^{2}}$.
 Hence, 6, 8 and 10 are Pythagorean triples.

Note: In this question, it is very important to know the following: the definition of Pythagoras theorem, a Pythagorean triplet consists of three positive integers a, b, and c, which satisfy the above condition ${{a}^{2}}={{b}^{2}}+{{c}^{2}}$, and that the longest side of a right angled triangle is its hypotenuse.