Problem 1

In a convex quadrilateral of area

32

, the sum of the lengths of two opposite sides and one diagonal is

16

. Determine all possible lengths of the other diagonal.

Step 5 of 7: Record equality conditions

In plain words

All slack has disappeared, so the two side segments are perpendicular to the diagonal.

AB+DC=8,\quad AB\perp BD,\quad DC\perp BD

Detailed analysis

Since $BD=8$ and $AB+BD+DC=16$ , we have $AB+DC=8$ . Equality in the altitude bounds gives $AB\perp BD$ and $DC\perp BD$ .