Question

Perimeter of a square is equal to the perimeter of an equilateral triangle whose one side is 12 cm long. The area of the square is:
(a) 144 sq. cm
(b) 81 sq. cm
(c) 64 sq. cm
(d) 100 sq. cm

Answer 1

Hint: Let us take the rough figures of the triangle and the square as follows.

We use the condition that all the sides of a square are equal and all sides of an equilateral triangle are equal. Also, the perimeter of any figure is given as the sum of the sides of the figure.
By using the above condition we find the side length of the square.
We have the formula of area of a square having the side length as \[a\] is given as
\[A={{a}^{2}}\]
Complete step by step answer:
We are given that the side length of an equilateral triangle as 12 cm
We know that all the sides of an equilateral triangle are equal
By suing the above condition to equilateral triangle \[\Delta PQR\] then we get
\[\Rightarrow PQ=QR=RP=12cm\]
We know that the perimeter of a figure is given by the sum of all sides of the figure.
By using the above condition we get the perimeter of triangle as
\[\begin{align}
  & \Rightarrow {{P}_{1}}=PQ+QR+RP \\
 & \Rightarrow {{P}_{1}}=12+12+12 \\
 & \Rightarrow {{P}_{1}}=36 \\
\end{align}\]
Now, let us assume that the side length of the square as \[a\]
We know that all the sides of a square are equal to each other.
By using the above condition to square ABCD we get
\[\Rightarrow AB=BC=CD=DA=a\]
We know that the perimeter of a figure is given by the sum of all sides of the figure.
By using the above condition we get the perimeter of square as
\[\begin{align}
  & \Rightarrow {{P}_{2}}=AB+BC+CD+DA \\
 & \Rightarrow {{P}_{2}}=a+a+a+a \\
 & \Rightarrow {{P}_{2}}=4a \\
\end{align}\]
We are given that the perimeter of the square is equal to the perimeter of the equilateral triangle.
By converting the above statement to mathematical equation we get
\[\begin{align}
  & \Rightarrow {{P}_{2}}={{P}_{1}} \\
 & \Rightarrow 4a=36 \\
 & \Rightarrow a=9 \\
\end{align}\]
Now, let us assume that the area of square as \[A\]
We know that the formula of area of square having the side length as \[a\] is given as
\[A={{a}^{2}}\]
By using the above formula we get the area of square as
\[\begin{align}
  & \Rightarrow A={{9}^{2}} \\
 & \Rightarrow A=81 \\
\end{align}\]
Therefore, we can conclude that the area of the given square as 81 sq. cm
So, option (b) is the correct answer.

Note:
Students may make mistakes in taking the condition that the perimeter of the square and the perimeter of the equilateral triangle are equal.
If the perimeters are equal for two different figures then the sum of the sides of the respective figures will be equal.
So, we get the equation as
\[\begin{align}
  & \Rightarrow {{P}_{2}}={{P}_{1}} \\
 & \Rightarrow 4a=36 \\
 & \Rightarrow a=9 \\
\end{align}\]
But students may make mistakes and assume that if the perimeters of two figures are equal then sides are equal. Then they take the equation as
\[\Rightarrow a=12\]
This is wrong because the number of sides of a triangle is 3 whereas the number of sides of the square is 4
So, there will be changes in the perimeter.