B4. Inside a rectangle $A B C D$ there is a point $P$ with $|A P|=6$, $|B P|=7$ and $|C P|=5$. How long is the line segment $D P$?

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B4. $2 \sqrt{3}$ Let the perpendicular projections of $P$ on the sides $A B$, $B C$, $C D$, and $D A$ be $Q$, $R$, $S$, and $T$ respectively. Then we have: $|A Q|^{2}+|Q P|^{2}=36$ and $|B Q|^{2}+|S P|^{2}=25$ (since $|B Q|=|C S|$), so $|A Q|^{2}+|Q P|^{2}+|B Q|^{2}+|S P|^{2}=61$. Furthermore, $|B Q|^{2}+|Q P|^{2}=49$. Therefore, $|D S|^{2}+|S P|^{2}=|A Q|^{2}+|S P|^{2}=61-49=12$ and $|D P|=\sqrt{12}=$ $2 \sqrt{3}$.

(c) Stichting Nederlandse Wiskunde Olympiade

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