Question

If diagonal of a square is 13cm, then find its side.

Answer 1

Hint:
We will use the Pythagoras theorem in the right triangle formed by the sides and diagonal of the square as: $\text{hypotenus}{\text{e}}^2\text{ = perpendicula}{\text{r}}^2\text{ + bas}{\text{e}}^2$ since all angles (vertex) of a square are of ${90}^{\circ }$.

Complete step by step solution:
We need to find the length of the side of the square.

Since all sides of the square are equal, let the sides be of length x cm each. And AC be the diagonal of the square whose length is 13cm.
We know that all the corner angles of a square are of ${90}^{\circ }$, therefore, we can use Pythagoras theorem given by: $\text{hypotenus}{\text{e}}^2\text{ = perpendicular}{\text{r}}^2\text{ + bas}{\text{e}}^2$
Using Pythagoras theorem in $△ABC$, we get
$$\begin{gathered} \Rightarrow \text{hypotenus}{\text{e}}^2\text{ = perpendicula}{\text{r}}^2\text{ + bas}{\text{e}}^2 \\ \Rightarrow {\left(AC\right)}^2={\left(AB\right)}^2+{\left(BC\right)}^2 \end{gathered}$$
Substituting the values of AC, AB and BC, we get
$$\begin{gathered} \Rightarrow {\left(13\right)}^2={\left(x\right)}^2+{\left(x\right)}^2 \\ \Rightarrow 169=2x^2 \\ \Rightarrow x^2={\displaystyle \frac{169}{2}} \\ \Rightarrow x=\sqrt{{\displaystyle \frac{169}{2}}}={\displaystyle \frac{13}{\sqrt{2}}}={\displaystyle \frac{6.5\times 2}{\sqrt{2}}}=6.5\sqrt{2}\text{cm} \end{gathered}$$

Therefore, the side of the square is of length $6.5\sqrt{2}\text{cm}$.


Note:
In this question, you can get confused in the calculation of x from the equation obtained after using the Pythagoras theorem. You can calculate it directly by taking the square root of ${\displaystyle \frac{169}{2}}=84.5$for the value of x instead of converting it into the form of $6.5\sqrt{2}$. You can also solve this question by directly applying the formula of relation between the diagonal of the square and its sides given as: Diagonal = $\sqrt{2}a$ , where a is the length of the side of the square, instead of using the Pythagoras theorem. This method will be helpful in MCQs to conserve time and get solutions quickly.