The problem asks to draw a square with a side length of 40 mm. using only a compass and ruler, and then to measure and dimension its diagonal.

Draw a line segment $\overline{AB}$ of length $4 \text{ cm}$. This step is correctly described in the image.

The description in the image for constructing a perpendicular line at point $A$ is unclear and contains errors. Here is a correct method:

• Extend the line segment $\overline{AB}$ beyond $A$ to a point $X$.
• Place the compass at $A$ and draw arcs of equal radius to intersect the line at two points, $P$ (on $\overline{AX}$) and $Q$ (on $\overline{AB}$).
• Open the compass to a radius greater than $AP$. Place the compass at $P$ and draw an arc above $A$.
• With the same radius, place the compass at $Q$ and draw an arc that intersects the previous arc. Label this intersection point $R$.
• Draw a straight line from $A$ through $R$. This line is perpendicular to $\overline{AB}$.

On the perpendicular line from $A$ (the line $\overline{AR}$), use the compass to mark a point $D$ such that $\overline{AD}$ is $4 \text{ cm}$ long.

• Place the compass at $D$ and draw an arc with a radius of $4 \text{ cm}$.
• Place the compass at $B$ and draw another arc with a radius of $4 \text{ cm}$.
• The intersection of these two arcs is point $C$.

Draw line segments $\overline{BC}$ and $\overline{CD}$ to complete the square $ABCD$.

For a square with side length $s$, the length of the diagonal $d$ can be found using the Pythagorean theorem: $d^2 = s^2 + s^2$.

To dimension it numerically: