24. $\triangle A B C$ is an equilateral triangle of side length 30 . Fold the triangle so that $A$ touches a point $X$ on $B C$. If $B X=6$, find the value of $k$, where $\sqrt{k}$ is the length of the crease obtained from folding.
24. Answer: 343 .

Apply the law of cosine on $\triangle X B Y$ and $\triangle X C Z$ respectively:
$$
\begin{array}{l}
p^{2}=6^{2}+(30-p)^{2}-6(30-p), \\
q^{2}=24^{2}+(30-q)^{2}-24(30-q) .
\end{array}
$$

Then $p=14$ and $q=21$. Applying the law of cosine in $\triangle Y X Z$ again to obtain
$$
k=r^{2}=p^{2}+q^{2}-p q=14^{2}+21^{2}-14 \cdot 21=343 .
$$