Question

A square piece of paper is folded as shown. if the area of the shaded triangle is \[225c{{m}^{2}}\], find the perimeter of the piece of paper when it is unfolded.

Answer 1

Hint: Since the area of triangle is half the area of square and adding both the shaded triangle will make a square .
So we can find the side of the square by the given area .
Then the side of the sheet will be twice the side of the obtained square and hence find the perimeter.

Formula used:
\[{{A}_{s}}={{a}^{2}}\]
Where \[{{A}_{s}}\] is the area of one of the squares in these four squares in the sheet.
\[a\]is the side of the square
\[P=4\times 2a\]
\[P\] is the perimeter of the sheet

Complete step by step solution:
Since both triangle is equal to each other and one triangle is just half of the quad square so total shaded area will be equal to the area of the one square
Let its side be \[a\]

Area of the shaded region \[=225c{{m}^{2}}\]
 Also, area of square is square of its side
\[\begin{align}
  & {{A}_{s}}={{a}^{2}} \\
 & \Rightarrow 225c{{m}^{2}}={{a}^{2}} \\
 & \Rightarrow a=15cm \\
\end{align}\]
Now side of the sheet is twice the side of this square
So:
 \[\begin{align}
  & \Rightarrow 2a=2\times 15cm \\
 & 2a=30cm \\
\end{align}\]
Now side of sheet \[=30cm\]
Perimeter of the sheet is four times the side of sheet
So:
 \[\begin{align}
  & P=4\times 2a \\
 & P=8\times 15cm \\
 & P=120cm \\
\end{align}\]
Perimeter of the unfolded sheet is \[120cm\].

Note: if we are finding the perimeter of the total sheet then take the whole length of the square that includes the side of the folded one as when it unfolded it's too form a similar square like others.
While calculating the side of the square twice the length of the square to get the side of the sheet.