Answer
Hypotenuse $=2\sqrt{6}$ inches.
Work Step by Step
The given lengths of the legs are
$a=\sqrt {10}-\sqrt2$ inches
$b=\sqrt {10}+\sqrt2$ inches
By using Pythagorean Theorem.
Hypotenuse $h=\sqrt{a^2+b^2}$.
Substitute all values.
$\Rightarrow h=\sqrt{(\sqrt {10}-\sqrt2)^2+(\sqrt {10}+\sqrt2)^2}$
Use special formulas
$(A-B)^2=A^2-2AB+B^2$
and $(A+B)^2=A^2+2AB+B^2$
$\Rightarrow h=\sqrt{(\sqrt {10})^2-2(\sqrt {10})(\sqrt2)+(\sqrt2)^2+(\sqrt {10})^2+2(\sqrt {10})(\sqrt2)+(\sqrt2)^2}$
Simplify.
$\Rightarrow h=\sqrt{10-2\sqrt {20}+2+10+2\sqrt {20}+2}$
$\Rightarrow h=\sqrt{24}$
$\Rightarrow h=\sqrt{4\cdot 6}$
$\Rightarrow h=2\sqrt{6}$ inches.
