Question

Write two Pythagorean triplets each having one of the numbers as $5$.

Answer 1

Determine the Pythagorean triple.

A Pythagorean triple consists of three positive integers $a,b,$ and $c$, such that $a^2+b^2=c^2$. Here, $a$ is the perpendicular, $b$ is the base and $c$ is the hypotenuse of the right-angled triangle.

Use the general formula of the Pythagorean triple.

$2mn,m^2–n^2,m^2+n^2$

Consider the first value:

$$\begin{gathered} m^2+n^2=5 \\ \Rightarrow m=2,n=1 \end{gathered}$$

Consider the second value:

$$\begin{gathered} 2mn=2\times 2\times 1 \\ \Rightarrow =4 \end{gathered}$$

Consider the third value:

$$\begin{gathered} m^2-n^2=2^2-1 \\ \Rightarrow =4-1 \\ \Rightarrow =3 \end{gathered}$$

Similarly, calculate the $(12,5,13)$ for $m=3,n=2$

Hence, the $\mathbf{3}\mathbf{,}\mathbf{}\mathbf{4}\mathbf{,}\mathbf{}\mathbf{5}\mathbf{}\mathbf{and}\mathbf{}\mathbf{12}\mathbf{,}\mathbf{}\mathbf{5}\mathbf{,}\mathbf{}\mathbf{13}$ are the two Pythagorean triplets