# Can we compute the area of a quadrilateral with one right angle when we only know the lengths of any three sides?

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:

1. Is my criticism valid or am I just being too proud because I got a question wrong?
2. Given my interpretation, DC is not a right angle, can this problem be solved?
• You know it is a right angle because it has a large "90" on it. Now we can argue they never said why it has a "90" on it and as I am a nitpick I would agree with you... but... I think you and I would lose in any court. – fleablood Apr 3 '19 at 2:45
• Not that angle, the one below it. – Robert Israel Apr 3 '19 at 2:46
• " even though an angle looks like an angle, it shouldn't be assumed" but it doesn't even look like a right angle. – fleablood Apr 3 '19 at 2:56
• Where did you find that test? Online IQ tests are generally untrustworthy even before we get to this quadrilateral problem. Many don't even bother ending in an IQ estimate. – J.G. Apr 3 '19 at 19:09
• yes for Cyclic quadrilaterals. Brahmagupta the Indian mathematician of the 7th century CE knew it. – R.C.Cowsik Apr 10 '19 at 5:45

You are right. The provided explanation is nonsensical. $$DC$$ cannot be assumed to be a right angle.

However, if you don't make that assumption, and take $$BC$$ as the only given right angle, the correct answer is "All four sides must be known."

The quadrilateral can be decomposed into two non-overlapping triangles. The first is a right angled triangle formed by sides $$B$$, $$C$$ and a hypotenuse, and its area is easy to determine. You can use Pythagoras' Theorem to find the hypotenuse of that right triangle formed by sides $$B$$ and $$C$$. That hypotenuse, together with sides $$A$$ and $$D$$ forms the other triangle. Its area can be computed using Heron's formula. Just sum the areas.

Note that I'm still making a tacit assumption that this is a convex quadrilateral. A concave quadrilateral is possible with the angle $$AD$$ being reflex, which is seemingly not representative of the diagram given, but is a possibility when one only has the information of a single right angle and four given sides. In the case of a concave quadrilateral, the area calculation that I detailed above would not be correct.

• For this reasoning to work, don't you also have to assume that the angle between A and D is obtuse? Without that restriction, there could be another quadrilateral with all the same side lengths but a different area. – David Z Apr 3 '19 at 9:06
• +1 for Heron's formula - I learned something new today – slebetman Apr 3 '19 at 9:31
• @DavidZ No further assumptions are needed. The shape may be different, but the area will be preserved. – Deepak Apr 3 '19 at 9:38
• @Deepak No, that's not correct. I took some time to double check. For example, see this image representing two quadrilaterals, both satisfying the conditions in the problem, both with the same side lengths, but having different areas - one is strictly smaller than the other. – David Z Apr 3 '19 at 10:02
• @Deepak Yes, that was exactly my point, that you can have a concave and convex quadrilateral with the same side lengths. Given that the question is about not assuming things that aren't explicitly stated, I think it may be worth at least mentioning that assumption. – David Z Apr 3 '19 at 10:41

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.

You are right: there is absolutely no indication that angle $$DC$$ is a right angle. If they wanted you to assume it was a right angle, they should have indicated that with another $$90$$. It really doesn't even look like a right angle (somebody had the bright idea of trying to render the picture in perspective, but we don't even know where the horizon is supposed to be).

• That's what I thought. It should explicitly state if any angles are right. However my second question remains, given DC is ambiguous, is this question solvable? I don't think there would be enough information to solve in this case. – Jack Apr 3 '19 at 2:52
• @JackO. See my answer. The correct answer would be "All sides must be known". – Deepak Apr 3 '19 at 3:11
• If we know all four lengths and assume no angle is more than 180, then I think there is only one quadrilateral so the area will be unique. I think. But you need all four. If you only three the fourth can be many lengths if the third one "swings". – fleablood Apr 3 '19 at 3:15
• @fleablood: No, in general there are infinitely many quadrilaterals with the same sides in the same order. See Bretschneider's formula for area of a quadrilateral given all sides and 2 opposite angles. – user21820 Apr 3 '19 at 12:04
• The rectangle appears to be drawn with "thickness", and if we assume that the thickness is perpendicular to the other sides, we might be able to figure out where the horizon is ... but I'm not sure that there is actually any consistent solution; the other two corners don't have thickness, which imposes constraints on the horizon (it must be such that the other "side" of the thickness for those corners is behind the corners) that may not be satisfiable. – Acccumulation Apr 3 '19 at 15:46