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(a) 8, 10, 6

(b) 15, 10, 5

(c) 63, 27, 3

(d) 80, 40, 20

Answer

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A Pythagorean Triplet, as the name suggests, consists such that the square of one of the numbers in the triplet is equal to the sum of the squares of the other two numbers. Pythagorean triplets are result of Pythagoras theorem of right angled triangles which state that the sum of the squares of lengths of the sides other than the hypotenuse in a right angled triangle is equal to the square of the length of the hypotenuse. With this relation, the biggest number in the triplet represents the hypotenuse and the other two numbers are sides of the right angled triangle.

Now, we shall consider the options one by one to check whether they are Pythagorean triplets or not and if they are, is one of the number 8 or not.

Let’s start with option (d) first.

The numbers in option (d) are 80, 40, 20.

The square of the biggest number 80 is 6400.

Square of 40 is 1600 and 20 is 400.

$ \begin{align}

& \Rightarrow LHS=6400 \\

& \Rightarrow RHS=1600+400 \\

& \Rightarrow RHS=2000 \\

& \Rightarrow LHS\ne RHS \\

\end{align} $

Thus, option (d) does not verify as it does not follow the basic condition to be Pythagorean triplet.

Now we shall see option (c).

The numbers in option (c) are 63, 27, 3.

The square of the biggest number 63 is 3969.

Square of 27 is 729 and 3 is 9.

$ \begin{align}

& \Rightarrow LHS=3969 \\

& \Rightarrow RHS=729+9 \\

& \Rightarrow RHS=738 \\

& \Rightarrow LHS\ne RHS \\

\end{align} $

Thus, option (c) does not verify as it does not follow the basic condition to be Pythagorean triplet.

The numbers in option (b) are 15, 10, 5.

The square of the biggest number 15 is 225.

Square of 10 is 100 and 5 is 25.

$ \begin{align}

& \Rightarrow LHS=225 \\

& \Rightarrow RHS=100+25 \\

& \Rightarrow RHS=125 \\

& \Rightarrow LHS\ne RHS \\

\end{align} $

Thus, option (b) does not verify as it does not follow the basic condition to be Pythagorean triplet.

This leaves us with option (a).

The numbers in option (a) are 8, 10, 6.

The square of the biggest number 10 is 100.

Square of 8 is 64 and 6 is 36.

$ \begin{align}

& \Rightarrow LHS=100 \\

& \Rightarrow RHS=64+36 \\

& \Rightarrow RHS=100 \\

& \Rightarrow LHS=RHS \\

\end{align} $

Thus, option (a) satisfies the condition and also contains 8.