+21 five points $a$, $b$, $c$, $d$, and $o$ lie on a flat field. $a$ is directly north of $o$, $b$ is directly west of $o$, $c$ is directly south of $o$, and $d$ is directly east of $o$. the distance between $c$ and $d$ is 140 m. a hot-air balloon is positioned in the air at $h$ directly above $o$. the balloon is held in place by four ropes $ha$, $hb$, $hc$, and $hd$. rope $hc$ has length 150 m and rope $hd$ has length 130 m. how high is the balloon above the field (that is, the length of $oh$)?
Let c = oc, d = od, and h = oh.
Since HC = 150, then h^2 + c^2 = 150^2 by the Pythagorean Theorem. Since HD=130, then h^2+d^2=130%2. Since CD=140, then c^2+d^2=140^2. Adding the first two equations, we obtain 2h^2+c^2+d^2=150^2+130^2.
Since c^2+d^2=140^2, then
\begin{align*} 2h^2 + 140^2 &= 150^2 + 130^2 \\ 2h^2 &= 150^2 + 130^2 - 140^2 \\ 2h^2 &= 2800 \\ h^2 &= 1400 \\ h &= \sqrt{1400} = \boxed{30\sqrt{11}}. \end{align*}
Therefore, OH = 30*sqrt(11).
