HMMT-Feb 2018 geo, Problem 2

Points $A, B, C, D$ are chosen in the plane such that segments $A B, B C, C D, D A$ have lengths $2,7,5$, 12 , respectively. Let $m$ be the minimum possible value of the length of segment $A C$ and let $M$ be the maximum possible value of the length of segment $A C$. What is the ordered pair $(m, M)$ ?

Solution: By the triangle inequality on triangle $A C D, A C+C D \geq A D$, or $A C \geq 7$. The minimum of 7 can be achieved when $A, C, D$ lie on a line in that order. By the triangle inequality on triangle $A B C$, $A B+B C \geq A C$, or $A C \leq 9$. The maximum of 9 can be achieved when $A, B, C$ lie on a line in that order. This gives the answer $(7,9)$. $\fbox{(7,9)}$.

Answer: (7,9)