Chapter 5 - Section 5.4 - The Pythagorean Theorem - Exercises - Page 241: 33

$UT=5\sqrt 5$

In $\triangle RST$ $RT^{2}=RS^{2}+ST^{2}$ $RT^{2}=6^{2}+8^{2}$ $RT^{2}=36+64$ $RT^{2}=100$ $RT=10$ now in $\triangle UTR$ $UT^{2}=RU^{2}-RT^{2}$ $UT^{2}=15^{2}-10^{2}$ $UT^{2}=225-100$ $UT^{2}=125$ $UT=\sqrt 125$ $UT=5\sqrt 5$