Polygon Coordinates given the Area and Center point

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Discussion Overview

The discussion revolves around the challenge of calculating the coordinates of a polygon, specifically a quadrilateral, given a known center point, area, and one known vertex (Point A), while the other vertices (Points B, C, and D) are unknown. The conversation explores potential methods and formulas for solving this problem, touching on geometric properties and assumptions about the shape of the polygon.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that the polygon is a quadrilateral, which could take various forms such as a square, rectangle, or trapezoid.
  • One participant proposes starting with a right triangle that includes the center point and Point A, and suggests that Point B could be chosen to create a triangle with one quarter of the total area.
  • Another participant questions the reasoning behind using one quarter of the area, noting that a polygon is not necessarily composed of equal triangles.
  • A participant emphasizes the importance of defining what is meant by the "center point," suggesting it may refer to the centroid, which could influence the solvability of the problem.
  • Some participants note that a quadrilateral has 8 degrees of freedom, and with only 5 values provided, additional information is necessary to solve the problem uniquely.
  • There is a discussion about whether the problem implies a specific type of polygon, with some suggesting it might indicate a square or rhombus, while others argue that the lack of specificity makes it difficult to infer the intended shape.
  • One participant introduces a mathematical approach involving vectors and determinants to express the area of the polygon, suggesting a method to find the unknown coordinates based on the known area.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the problem statement regarding the type of polygon and the necessary conditions for a unique solution. There is no consensus on whether the problem can be resolved without additional information or whether it implies a specific geometric figure.

Contextual Notes

Participants highlight limitations related to the assumptions about the center point and the nature of the polygon, as well as the mathematical steps required to arrive at a solution. The discussion reflects uncertainty regarding the uniqueness of the solution based on the given parameters.

PhysicsInept
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TL;DR
Polygon
I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following
- Center point is known
- area is known
- Point A is known
- Points B, C, and D are UNKNOWN

I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this was even covered in high school math!
Thanks
 
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So the polygon has 4 points meaning it's a quadrilateral of some sort like a square, rectangle, rhombus, parallelogram, trapezoid or some arbitrary shape.

One could start with a right triangle contain the center point and point A choose B such that the triangle has one quarter of the area given.
 
Hmmm okay so bear with me as I try to understand ..
So I would use the Center point as Point A of a right triangle, then I know Point B already, and I can calculate the last Point since I have area. Why would the area be 1/4 of what I have been given? A polygon isn’t 4 equal triangles right?
 
PhysicsInept said:
I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following
- Center point is known
As you only mention four points we need to assume the polygon is a quadrilateral - this is important. It is also important to define what you mean by the "center point": if you mean what is usually called the centroid (also sometimes geometric centre/center or even barycentre/barycenter) then yes this should be solvable with some work; it will probably involve Varignon's parallelogram.

jedishrfu said:
One could start with a right triangle contain the center point and point A choose B such that the triangle has one quarter of the area given.
I assume you mean that the "centre point" is the right angled vertex of the triangle? That will construct a quadrilateral with the "centre point" at the intersection of the diagonals which is not in general the centroid.
 
Yes I didn't want to solve it for the OP.
 
A polygon with four sides has 8 degrees of freedom (e.g. the 4 corners), we only have 5 values given. This can only be solved if we know it's a special type of polygon.
 
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mfb said:
A polygon with four sides has 8 degrees of freedom (e.g. the 4 corners), we only have 5 values given. This can only be solved if we know it's a special type of polygon.
It can only be solved uniquely if we have more information, but I'm not sure if uniqueness is required (the OP doesn't ask for "the polygon").
 
If a question asks for the coordinates of points without saying anything else it's usually a unique solution (only dependent on given parameters). A general polygon doesn't have a single unambiguous center either, that makes me think there is some information missing in the first post. If you require some special shapes and define their center then you can find solutions of course.
 
To my mind it implies a square or a rhombus.
 
  • #10
PhysicsInept said:
I’m wondering if there is a formula for calculating the coordinate points of a polygon

jedishrfu said:
To my mind it implies a square or a rhombus.
I don't agree, based on the OP. If someone asks about a polygon, without additional information, it's a stretch to infer that a more specific geometric figure is meant. Even if we take the mention of points A, B, C, and D to mean that these are vertices, and that there are no other vertices, the figure could still be a rectangle, trapezoid, or just an irregular quadrilateral.
Seeing the exact problem statement would clear up these uncertainties.
 
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  • #11
If you take a polygon with 4 points, there is always an inner diagonal. Take as origin one of the endpoints of the diagonal and name the 3 other endpoints u, v and w as vectors where v is the other endpoint of the inner diagonal.

Then the formula for the surface of the polygon equals ##|(Det(u\: v) + Det(v\: w)/2|##

Thus if this area a is known, you can take angle phi and take 2 matrices from ##SL_2(R)## where the 1st times ##cos(\phi)## has as 2nd column the 1st column of the 2nd matrix times ##sin(\phi)##. These matrices multiplied with ##a\sqrt(1/2)cos(\phi)## or ##sin(\phi)## for the 2nd have as columns the vectors we wanted to find.
 
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