Question

The diagonal of a square is $10$ cm long. How do you find the length of each side of the square?

Answer 1

Hint: We first assume the length of the side of the square as x cm. Then, applying the Pythagoras theorem to the triangle formed by the diagonal and two sides, we get the value of x.

Complete step-by-step solution:
A quadrilateral is a polygon with four sides. A square or a rhombus is a quadrilateral with all the four sides equal. The only difference between a square and a rhombus is that all the sides of a square are perpendicular to their adjacent sides, but in a rhombus it is not. This is what differentiates a square from the rest of the quadrilaterals.
In this problem, it is given that for a particular square, the length of its diagonal is $10$ cm. Now, if we consider the length of one side of the square as x cm, then all four sides will be equal to x cm. We also know that the adjacent sides of a square intersect at right angles. Thus, we can perfectly apply the Pythagoras theorem for a right-angled triangle which states that for a right-angled triangle with a and b as its sides and c as its hypotenuse, the relation between them can be stated as ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$ . Comparing it with the square, the diagonal behaves as the hypotenuse while the sides of the square behave as the sides of the right-angled triangle.

Applying the Pythagoras theorem, we get,
$\begin{align}
  & {{10}^{2}}={{x}^{2}}+{{x}^{2}}=2{{x}^{2}} \\
 & \Rightarrow {{x}^{2}}=50 \\
 & \Rightarrow x=\sqrt{50}=5\sqrt{2}cm \\
\end{align}$
Therefore, we can conclude that the sides of the square are $5\sqrt{2}cm$ each.

Note: The Pythagoras theorem must be applied correctly, taking the square of all the three terms. The problem can also be solved using a shortcut formula. If the side of a square is a, then the length of its diagonal will be $a\sqrt{2}$ . Equating $a\sqrt{2}$ to $10$ cm, we get the value of a.