## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$10 \text{ and } 5\sqrt{3}\approx8.660$
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.