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In quadrilateral ABCD, we have AB = 3, BC = 6, CD = $$9$$, and DA = $$12$$.
If the length of diagonal AC is an integer, what are all the possible values for AC? Explain your answer in complete sentences.

May 13, 2022

Triangle inequality implies $$\begin{cases} 3 + 6 > AC\\ 3 + AC > 6\\ 6 + AC > 3\\ 9 + AC > 12\\ 9 + 12 > AC\\ 12 + AC > 9 \end{cases}$$. If you list the integers satisfying all 6 inequalities, the possible values would be 4, 5, 6, 7, 8.