The cube $ABCDEFGH$ has an edge of the length $a$. Bob was tasked with calculating the length of the space diagonal u of the given cube. He proceeded as follows:
(1) First, Bob made a sketch of the cube.
(2) Then, he decided to use the Pythagorean theorem to calculate the length of the diagonal: $$ u=\sqrt{ |AB|^2 + |BG|^2} $$ (3) Next, he substituted the lengths of the edges into the equation above and simplified to get the result: $$ \begin{align} u &= \sqrt{a^2 + a^2} \cr u &=a\sqrt2 \end{align} $$ Decide whether Bob made a mistake. If so, determine where.
Bob did not make a mistake, everything is correct.
Bob made a mistake in step (1). The triangle $ABG$ cannot be used to calculate the length of the space diagonal. He should have used the hypotenuse of the triangle $ABF$.
Bob made a mistake in step (2). He cannot use the Pythagorean theorem to calculate the hypotenuse of the triangle $ABG$ since the triangle is not right-angled.
Bob made a mistake in step (3). He incorrectly substituted into the formula for calculating the length of the space diagonal. Therefore, the result is incorrect.
Bob made the mistake in step (3) in determining the length of the side diagonal $BG$. Its true length is $a\sqrt2$. The correct calculation of the length of the space diagonal is: $$ \begin{align} u &=\sqrt{a^2 + \left(a\sqrt{2}\right)^2} \cr u &=a\sqrt3 \end{align} $$
