I am given three sides of length 4.5 and one more side of length 4.2 (dimensionless entities). Also the area is given as 19.575. Now I have been given the task to construct a quadrilateral with these figures. I don't know whether it is trivial or complex. Since the type of quadrilateral is not specified, I'm concerned whether such a polygon can be constructed (I tried considering trapezium), and if possible, whether it is unique! And when unique, how to construct. I need help with this. Any online source for construction is also welcome. The main goal is to know angles between the sides. Thanks in advance.
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Check out https://math.stackexchange.com/questions/873820/what-is-the-maximum-area-of-a-quadrilateral-with-sides-of-length-a-b-c-d-in-seq to see whether this quadrilateral exists. – player3236 Nov 10 '20 at 02:54
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I am going to assume that the quadrilateral is constructible.
The first thing to do is check out Heron's formula from here.
Draw a main diagonal and assume its length is $x$. Then you have two triangles. You can compute the area of both triangles using Heron's formula, as a function of $x$. Since the quadrilateral's area is given, you know the area of the two triangles formed by the main diagonal.
Therefore, you can solve for the length of the main diagonal. That is, you can solve for $x$. Once this is done, you can solve for two of the angles via the Law of Cosines.
Then, rinse and repeat, to get the length of the other main diagonal, and consequently the other two angles.
Addendum
Never having done this, it is unclear to me whether the computation of each diagonal will yield a unique value. From my perspective, that is actually irrelevant because when you identify each of the 4 angles that correspond to a pair of (possible diagonal lengths), you have the constraint that the 4 angles must sum to $360^{\circ}.$
user2661923
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I'm going to assume that it isn't. You might want to check out what I rendered. – Oscar Lanzi Apr 28 '21 at 15:19
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@OscarLanzi "I am going to assume that the quadrilateral is constructible." The point of this sentence is meta-cheating. That is, I am assuming that this is an assigned math problem, rather than (for example) a real life architectural problem. So, I side-stepped the whole existence issue by entering the mind of the problem composer. That is, the problem is difficult enough without throwing a trick question at the student, such as knowingly specifying an unconstructible quadrilateral. – user2661923 Apr 28 '21 at 16:27
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@OscarLanzi Notice that I am ducking the following question: suppose that you are assigned a specification that involves a non-constructible quadrilateral. Although I could be mistaken, my instinct is, that if you follow the procedure outlined in my answer, you will be forced to abort. That is, I am blindly speculating that at some point, either in solving for one of the diagonals, or solving for one of the angles, you will be presented with a math equation that demonstrably has no solution. Assuming that my blind speculation is true, then the procedure in my answer remains valid. – user2661923 Apr 28 '21 at 16:30
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Maybe the point of the problem never was to construct a quadrilateral but to have the student check for existence before investing a heavier time input to find the solution. – Oscar Lanzi Apr 29 '21 at 12:06
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Before breaking out your straightedge and compasses, you might want to read the story of the Emperor's new clothes. You need to check that the solution, like the clothes, is really there. Read on to discover the naked truth.
Bounds of Decency
Given a set of side lengths for any polygon, the maximum possible area is obtained by setting up the angles so the polygon is inscribed in a circle. With a quadrilateral having three congruent sides that would, of course, be an isosceles trapezoid. Two of the three congruent sides are the legs of the trapezoid, the remaining congruent side and the fourth side are the bases.
Thus consider a trapezoid with bases $4.2$ and $4.5$ and both legs $4.5$. Its altitude is then
$\sqrt{4.5^2-[(4.5-4.2)/2]^2}=\sqrt{20.2275}$
The area, which is the maximum possible area for our quadrilateral, is then half the sum of bases times this altitude:
$S_{max}=4.35\sqrt{20.2275}$
$=\color{blue}{19.564...<19.575}$
The Emperor, in fact, is wearing no clothes. The construction is impossible.
Oscar Lanzi
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