Answer
$10 \text{ and } 5\sqrt{3}\approx8.660$
Work Step by Step
The side opposite the $30^o$ angle of a right triangle measures half as long as the hypotenuse. Hence, the hypotenuse is $10.$
Let the right triangle have $a$ and $b$ as the legs and $c$ as the hypotenuse. Using $a^2+b^2=c^2$ or the Pythagorean Theorem, with $
c=10
$ and $
a=5
,$ then
\begin{array}{l}\require{cancel}
a^2+b^2=c^2
\\\\
5^2+b^2=10^2
\\\\
25+b^2=100
\\\\
b^2=100-25
\\\\
b^2=75
\\\\
b=\sqrt{75}
\\\\
b=\sqrt{25\cdot3}
\\\\
b=\sqrt{(5)^2\cdot3}
\\\\
b=5\sqrt{3}
.\end{array}
Hence, the missing measurements are $
10 \text{ and } 5\sqrt{3}\approx8.660
$ units long.
