TITLE: Can we compute the area of a quadrilateral with one right angle when we only know the lengths of any three sides?
QUESTION [22 upvotes]: I took an IQ test for fun recently, but I take issue with the answer to one of the questions. Here's the question: 

My issue is that the explanation assumes angle DC is a right angle. Given that assumption, I can see the quadrilateral is indeed a rectangle and a right triangle and can follow their explanation. However, (from what I remember my high school geometry teacher telling me) even though an angle looks like a right angle, it shouldn't be assumed unless it is explicitly stated or you can prove it. To explain what I mean, if DC isn't a right angle and we exacerbated that difference, it would look like the following:
 
Thus, even being given A, B, C and D it seems like the area could not be calculated.
So my question is twofold:

Is my criticism valid or am I just being too proud because I got a question wrong?
Given my interpretation, DC is not a right angle, can this problem be solved?

REPLY [16 votes]: You are correct that the given solution is wrong. Worse still, even if you know that the angles between BC and CD are both right-angles, the purported answer is still wrong! This is because if you're given the lengths of A,B,C, it still does not uniquely determine D because we are not told that the angle between AB is less than $90°$.
In general, it is not enough even if you have all four side lengths. For example, consider convex quadrilateral $PQRS$ such that $PQ = 6$ and $QR = 9$ and $RS = 7$ and $SP = 8$. It is possible that $P,R$ are slightly less than $15$ apart, making $PQRS$ a very skinny quadrilateral whose area can be made arbitrarily close to zero. Alternatively, moving $P,R$ to a distance of $10$ makes $PQRS$ rather square-like with area clearly more than $48$. Bretschneider's formula gives the area for an arbitrary quadrilateral, and you can see from it as well that fixing the side lengths is not enough to determine the area, since it also varies with the sum of two opposite angles.