Question

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

Answer 1

Hint: 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.