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.
Constructing a Quadrilateral
geometric-constructionquadrilateral
Related Solutions
Given two quadrilaterals $ABCD$ and $EFGH$, with side lengths equal, (i.e. $d(A,B)=d(E,F)$, etc.) then $ABCD \cong EFGH$ if and only if the diagonals have equal length: $d(A,C)=d(E,G)$ and $d(B,D)=d(F,H)$, where $d$ is the distance between the points.
The forward implication is obvious. To see the converse, we first notice that the diagonal's length being equal gives us a few congruent triangles. Namely, $\triangle ABC \cong \triangle EFG$, $\triangle ACD \cong \triangle EGH$ from $d(A,C)=d(E,G)$, by the good old SSS property.
Using only the one diagonals being equal, we would have two possible quadrilaterals from these facts. They come from reflecting one of the triangles over the diagonal.
Next, we see that we are given the distance between $D$ and $B$. Therefore, we are forced in which of these two options must be our congruent quadrilateral. The only way that both quadrilaterals could be acceptable is if $B$ is the proper distance from $D$ in both. But this will force $B$ onto the diagonal between $A$ and $C$, which contradicts the definition of a polygon (giving you a triangle instead), and finishes the proof.
I hope this was helpful. I believe a slight variation of this will work if you know the length of one diagonal, say $d(A,C)$ and the angle $\angle BCD$ or $\angle BAD$. It would seem reasonable that the same will hold if you know area, the four sides and a diagonal also. The fact that you have two possibilities right away from one triangle gives you most of the information. Also, it is pretty clear that the reflection over one of the diagonals will always produce one convex quadrilateral and one not.
Draw circle $c$ of center $C$ and radius $CA$, meeting side $CB$ at $D$ (I'm assuming $AC<BC$).
Draw circle $b$ of center $B$ and radius $BD$, meeting side $AB$ at $E$.
Draw line $ED$: its second intersection with circle $c$ is point $X$ such that $CX=CA$ (obvious) and $CX\parallel AB$ (because triangles $CDX$ and $BDE$ have the same angles).
Draw line $AX$, meeting side $BC$ at $X'$, which is a vertex of the rhombus.
Draw circle $d$ of center $X'$ and radius $X'D$, meeting line $ED$ at $F$; line $X'F$ is parallel to $AB$, because triangles $X'DF$ and $BED$ have the same angles.
Draw line $AD$, meeting circle $d$ at $G$; line $X'G$ is parallel to $AC$, because triangles $X'DG$ and $CDA$ have the same angles.
- Draw lines $X'F$ and $X'G$ to complete the rhombus.
Best Answer
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.