Question

Use the square shown. Find the exact length $\mathrm{x}$ of a diagonal. Is this number a rational number or an irrational number? Explain.

Answer 1

Given figure:

The given figure is square and the triangle so formed by the diagonal of the square is right angled triangle.

To find $\mathrm{x}$, use Pythagoras theorem

${\mathrm{a}}^2+{\mathrm{b}}^2={\mathrm{c}}^2$

Given $\mathrm{a}=1$ units, $\mathrm{b}=1$ units and $\mathrm{c}=\mathrm{x}$ units

Now,

$$\begin{gathered} {\left(1\right)}^2+{\left(1\right)}^2={\mathrm{x}}^2 \\ \Rightarrow 1+1={\mathrm{x}}^2 \\ \Rightarrow {\mathrm{x}}^2=2 \\ \Rightarrow \mathrm{x}=\sqrt{2} \end{gathered}$$

The value of $\mathrm{x}$ is $\sqrt{2}$ which is an irrational number as square root of any prime number is irrational number.

Hence, the value of $\mathrm{x}$ is $\sqrt{2}$ which is an irrational number.