Question

Find the perimeter of a rectangle whose one side measure and the diagonal 29 m is
A. \[82m\]
B. \[78m\]
C. \[98m\]
D. \[88m\]

Answer 1

Hint:
Here we need to find the perimeter of the rectangle given diagonal and given one side. We will first consider the triangle inside the rectangle and then we will apply the Pythagoras theorem in that. From there, we will get the length of the other side of the rectangle. Then at last, we will apply the formula of perimeter of the rectangle to get the required answer.

Formula used:
Perimeter of rectangle \[ = 2 \times \] (Length \[ + \] Breadth)

Complete step by step solution:
Here we need to find the perimeter of the rectangle given diagonal and given one side.
It is given that the one side of the rectangle is \[20m\] and the length of the diagonal is \[29m\].
We will first draw the rectangle \[ABCD\] along with its one diagonal.

Now, we will consider the right angled triangle \[ABC\] and we will apply the Pythagoras theorem.
\[A{C^2} = A{B^2} + B{C^2}\]
Now, we will substitute the value of the sides and the diagonals, we get
\[ \Rightarrow {29^2} = {20^2} + B{C^2}\]
On applying the exponent on the base, we get
\[ \Rightarrow 841 = 400 + B{C^2}\]
Now, we will subtract 400 from both sides.
\[\begin{array}{l} \Rightarrow 841 - 400 = 400 + B{C^2} - 400\\ \Rightarrow 441 = B{C^2}\end{array}\]
On taking square roots, we get
\[\begin{array}{l} \Rightarrow \sqrt {441} = \sqrt {B{C^2}} \\ \Rightarrow 21 = BC\end{array}\]
Therefore, the value of the other side of the rectangle is equal to \[21m\].
Now, we will find the perimeter of the rectangle.
On substituting the value of length and breadth in the formula Perimeter of rectangle \[ = 2 \times \] (Length \[ + \] Breadth), we get
Perimeter of rectangle \[ = 2 \times \left( {20 + 21} \right)\]
On adding the terms, we get
\[ \Rightarrow \]Perimeter of rectangle \[ = 2 \times 41\]
On multiplying the numbers, we get
\[ \Rightarrow \]Perimeter of rectangle \[ = 82m\]

Hence, the correct option is option A.

Note:
Here we have obtained the perimeter of the rectangle which we know is equal to twice the sum of the length and breadth of the rectangle. Here we have used the Pythagoras theorem which states that the square of the longest side of the right angled triangle i.e. hypotenuse is equal to the sum of the square of the other two sides of the triangle.