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 6 of 7: Compute the other diagonal

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In plain words

Coordinates expose a right triangle whose legs are the diagonal and the combined height.

AC^2=BD^2+(AB+DC)^2=8^2+8^2=128

Detailed analysis

Put $B=(0,0)$ and $D=(8,0)$ . Convexity places $A,C$ on opposite sides of $BD$ , so the vertical separation is $AB+DC=8$ . Therefore $AC^2=BD^2+(AB+DC)^2=8^2+8^2=128$ .