HMMT February

Geometry Difficulty 5.7 AIME, harder Prove it United States

Problem:

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

Solution

Solution:

By the triangle inequality on triangle $ACD$ , $AC + CD \geq AD$ , or $AC \geq 7$ . The minimum of $7$ can be achieved when $A$ , $C$ , $D$ lie on a line in that order.

By the triangle inequality on triangle $ABC$ , $AB + BC \geq AC$ , or $AC \leq 9$ . The maximum of $9$ can be achieved when $A$ , $B$ , $C$ lie on a line in that order.

This gives the answer $(7, 9)$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.