Two poles of heights $6 \, m$ and $11 \, m$ stand on a plane ground. If the distance between the feet of the poles is $12 \, m$,find the distance between their tops.

(13 M) Let $CD$ and $AB$ be the poles of height $11 \, m$ and $6 \, m$ respectively.
Draw a line segment $AP$ parallel to $BD$ such that $P$ lies on $CD$. Then $AP = BD = 12 \, m$ and $PD = AB = 6 \, m$.
Now,$CP = CD - PD = 11 \, m - 6 \, m = 5 \, m$.
In the right-angled triangle $\triangle APC$,by applying the Pythagoras theorem:
$AC^2 = AP^2 + CP^2$
$AC^2 = (12 \, m)^2 + (5 \, m)^2$
$AC^2 = 144 \, m^2 + 25 \, m^2 = 169 \, m^2$
$AC = \sqrt{169} \, m = 13 \, m$.
Therefore,the distance between their tops is $13 \, m$.