Question

A. Look at the diagram of the two squares shown below.

Answer 1

Squares 1 and 2 have the same area because they're both (a+b) on a side.

Each of these squares is covered by four identical right triangle tiles, legs a,b, hypotenuse c.  

In the first picture we see the uncovered part of the square, not covered by triangular tiles, is two squares, area $a^2+b^2$.

In the second picture the uncovered part of the square is a smaller square, area $c^2$.

We just moved the tiles around on the square, so the uncovered part is the same in both cases.  So

$a^2 + b^2 = c^2$

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The rectangular prism has a bottom rectangular base 6 by 8.  So  the diagonal is $\sqrt{6^2+8^2}=\sqrt{100}=10$.

The diagonal and the 7 cm side make a right triangle whose hypotenuse is the diagonal of the rectangular prism we seek.

$\sqrt{10^2 + 7^2} = \sqrt{149}$

Answer: √149