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A figure consists of 5 equal squares in the form of a cross. Please show how to divide it with two straight cuts into 4 equal pieces which will fit together to form a square.

A MSE told me I need to cut it from the vertices but how?

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Here is a good way of seeing how this dissection comes about.

The cross shape tiles the plane in a regular way. If you pick any point inside a cross, and mark the same point in all the crosses of the tiling, you get a grid of points that can be connected to form a grid of squares. Those grid-lines split up the crosses into pieces which also form the squares in that grid.
enter image description here
If the grid intersection lies anywhere in the middle square of the cross, then there will only be four pieces. If you put the grid intersection too far into one of the arms of the cross, then you will get five or even six pieces.
You can let the grid lines go exactly through the vertices, but I deliberately did not do so in my picture to illustrate the fact that it is a more general solution.
This only way to make the four pieces have the same shape is to put the grid intersection in the centre of the cross. The cuts then go through the midpoint of the sides of the arms of the cross, not through the vertices.

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Like this (not like the commented link):

enter image description here

Curiously there is

The same cross inside but tilted.

A different arrangement makes two squares:

enter image description here

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