Question

The perimeter of a square is $(4x + 20)$cm. what will be the length of its diagonal $?$

Answer 1

First we have to know that a square is a rectangle in which two adjacent sides have equal length (The length each side of a square is equal). First find the length of the side of a given square using the perimeter formula. Then to find the length of the diagonal, we need to multiply the length of one side by the square root of $2$.

Complete step by step answer:
Let $a$ and $P$be the length of each side and the perimeter of the given square respectively. Also let $d$ be the length of the diagonal of a square.
Then the perimeter of the square with each side $a$ $= 4a$
i.e., $P = 4a$-------(1)

Since given the perimeter of a square is equal to $(4x + 20)$cm i.e., $P = (4x + 20)$, then the equation (1) becomes
$4a = (4x + 20)$------(2)
Dividing the both sides of the equation (2) by $4$, we get
$a = (x + 5)$----(3)
We know that if $a$ is the side of a square, then the length of the diagonal of a square $= \sqrt 2 a$
i.e., $d = \sqrt 2 a$-----(4)
Substituting the value of $a$ from the equation (3) in the equation (4), we get
$d = \sqrt 2 (x + 5)$
Hence, the length of the diagonal of a square with perimeter $(4x + 20)$cm $= \sqrt 2 (x + 5)$cm.

Note:
A square is a regular quadrilateral and its diagonals cross in ${90^o}$ angle. Hence diagonals of a square are perpendicular to each other. Also note that the perimeter of a given is the length of the outline of a given shape. Hence to find the perimeter of a rectangle, square, or triangle you have to add the lengths of all the sides.