[Math] Cutting a cake into $2$ pieces of equal area

Question

You have a large rectangular cake, and someone cuts out a smaller rectangular piece from the middle of the cake at a random size, angle and position in the cake (see the picture below). Without knowing the dimensions of either rectangle, using one straight (vertical) cut, how can you cut the cake into two pieces of equal area?

Here is my attempt at the solution. I assume that all the dimensions are known.

Area of large rectangular cake $=ab\text{ }(a,b\in \mathbb{R}^+,a>b)$

Area of smaller rectangular piece $=cd\text{ }(c\in [0,a],d\in [0,b],c>d)$

Total area $=ab-cd$

Area right of vertical cut $=mb$ (we need to determine $m$)

Area left of vertical cut $=(a-m)b-cd$

Equate both expressions: $mb=(a-m)b-cd$

$$cd=(a-2m)b$$

Therefore

$$m=\frac{1}{2}\left(a-\frac{cd}{b}\right)$$

Is there a way to answer the question without assuming we know the dimensions.

Answer 1

You just need to find the center of each rectangle.