How to find the perimeter of a piece of paper which has been cut?

Question

The problem is as follows:

From a rectangular piece of paper, $4$ straight cuts have been made. These cuts are parallel to the diagonals of the rectangle. After making the cuts, the 4 pieces are removed. The sum of the lengths of the four cuts made is $\textrm{80 cm}$. Find the perimeter of the piece of paper that is obtained.

The alternatives given in my book are as follows:

$\begin{array}{ll} 1.&\textrm{292 cm}\\ 2.&\textrm{248 cm}\\ 3.&\textrm{276 cm}\\ 4.&\textrm{284 cm}\\ \end{array}$

For this particular situation. I'm stuck. Does it exist a way to get this perimeter?. The reason for why I'm stuck is that it isn't given the lengths of the corners of the paper neither full sides of the chunks which has been removed. Does it exist to get those, can this problem be solved?. Help please?

Answer 1

Note that $$\text{New perimeter} = \text{Old perimeter}-\text{Lengths of corners}+\text{Lengths of new edges}$$ Since the cuts are parallel to the diagonal of the rectangle, and one of the angle is $90^\circ$, by AA rule, every cut triangle is similar to the triangle formed by cutting the rectangle along one of its diagonals. So the sides of the cut triangles are in the ratio $3:4:5$ (as shown in the following image). So the new perimeter is $$2(60+80)-3(x+y+z+w)-4(x+y+z+w)+5(x+y+z+w)$$ Using the constraint that $$5(x+y+z+w)=80$$ and putting in above expression, we get, the new perimeter to be $$280-2\times\dfrac{80}5=\boxed{248\ \mathrm{cm}}$$

Answer 2

Recognize that if the cuts are parallel to the diagonals, then the side lengths of the triangles that have been cut out are in proportion to a $3-4-5$ right triangle.

When a cut is made, a length equal to the sum of the two legs of the right triangle will have been taken out of the perimeter, but the length equal to the hypotenuse will be added (i.e. for every $5cm$ cut, $7cm$ gets taken out of the perimeter, but $5cm$ will be added to the perimeter, giving a net loss of $2cm$ per $5cm$ cut).

Since we started with a perimeter of $280cm$, and $80cm$ of cuts have been made, this gives us a total loss of $32cm$, resulting in a perimeter of $248cm$.