In quadrilateral $ABCD$, we have $AB=3,$ $BC=6,$ $CD=4,$ and $DA=4$.

If the length of diagonal $AC$ is an integer, what are all the possible values for $AC$?

The triangle inequality applies.

In order for ACD to be a triangle, the length of AC must lie between CD-DA=0 and CD+DA=8.

In order for ABD to be a triangle, the length of AC must lie between BC-AB=3 and BC+AB=9.

The values common to both these restrictions are numbers between 3 and 8. Assuming we don't want the diagonal to be coincident with any sides, its integer length will be one of ...
{4, 5, 6, 7}

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In quadrilateral $ABCD$, we have $AB=3,$ $BC=6,$ $CD=4,$ and $DA=4$. If the length of diagonal $AC$ is an integer, what are all the possible values for $AC$?

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In quadrilateral $ABCD$, we have $AB=3,$ $BC=6,$ $CD=4,$ and $DA=4$.

If the length of diagonal $AC$ is an integer, what are all the possible values for $AC$?