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 7 of 7: Conclude and verify existence

Previous step Next step

In plain words

The answer is unique even though the quadrilateral itself has many shapes.

AC=8\sqrt2

Detailed analysis

Thus $AC=8\sqrt2$ . Conversely, any $AB,DC>0$ with $AB+DC=8$ , placed perpendicularly on opposite sides of an $8$ -segment $BD$ , gives a convex quadrilateral of area $32$ , so the value is attainable and unique.