Question

The figure below has been obtained by folding a rectangle. The total {visible area of the folded figure is $144\ \text{m}^2$. Had the rectangle not been folded, the current overlapping part would have been a {square}. What would have been the total area of the original (unfolded) rectangle?}

• A. 128 square meters
• B. 154 square meters
• C. 162 square meters
• D. 172 square meters
• E. None of the above

Hint:

For folding problems: \;Original area $=$ Visible area $+$ Overlap area. If the overlap is a square and the fold shows its diagonal, use the $45^\circ$ relation to get the square’s side.

Answer 1

The Correct Option is C

Step 1: Understand what “visible area” means after folding.
When a sheet is folded so that one part overlaps another, the region of overlap is made of {two coincident layers} but, geometrically, it occupies the area of the overlap just once.
Therefore, \[ \text{(area of original rectangle)} \;=\; \text{(visible area after fold)} \;+\; \text{(area of overlap)}. \] Step 2: Identify the overlap.
The problem states that the overlapped portion (when unfolded) is a {square}. In such a fold, the visible slanted edge is the {diagonal} of that square, making the two perpendicular offsets equal. From the given measures in the figure, these equal offsets turn out to be \(3\ \text{m}\), so the square’s side is \[ s \;=\; 3\sqrt{2}\ \text{m} \quad\Rightarrow\quad \text{overlap area }= s^{2}=18\ \text{m}^2. \] Step 3: Compute the original area.
\[ \text{Original area} \;=\; \text{Visible area} \;+\; \text{Overlap area} \;=\; 144 \;+\; 18 \;=\; \boxed{162\ \text{m}^2}. \] \[ \boxed{\text{Total area of the original rectangle }= 162\ \text{square meters}.} \]