What is the formula to find the one unknown vertex's coordinates?

Question

For a project, I need the following formulae. Kindly help me:

1) Triangle with Cartesian coordinates:

I know the coordinates of two vertices, the lengths of all the 3 sides and also the area of a triangle. The triangle is a scalene triangle and is not touching the x or y axes in anyway, and none of the sides of it are parallel or perpendicular to the x or y axes.

I want to know the formula for calculating the x and y coordinates of the third vertex.

Kindly help me.

2) Quadrilateral with Cartesian coordinates:

I know the coordinates of three vertices, the lengths of all the 4 sides and also the area of a quadrilateral. The quadrilateral is not touching the x or y axes in anyway, and none of the sides of it are parallel or perpendicular to the x or y axes.

I want to know the formula for calculating the x and y coordinates of the fourth vertex.

Kindly help me.

Thank you very much in advance.

Best regards,
D Rajakumar
Bangalore, India

Dear Mr Bramblyspam, Basically the Triangle as well as the Quadrilateral I have are placed randomly in one of the quardrants of the cartesian coordinate system. That is what I mean when I mention that they are not touching the x or y axes in anyway, and none of the sides are parallel or perpendicular to the x or y axes.

I hope that helps. Kindly provide me with an accurate formula for x and y of the one unknown vertex please. I appreciate your valuable help. Thanks very much.

Answer 1

Hello D Rajkumar,
There is a lot of redundnacy in 1st question
if you know the co-ordinates of A(x1,y1),B(x2,y2) then AB is known ir computable. Now if distance from C(say (x,y)) is known

AC^2=(x-x1)^2+(y-y1)^2
BC^2 =(x-x2)^2+(y-y2)^2
using the above 2 equations one can compute (X,y) there shall be 2 values one on each side.
If area is known then find AB and from this the distance then we have 2 lines || to AB for the point C.
For the quadrileteral similarly 2 locations for D can be found and from the area exact point for D can be found.

Answer 2

1.
Insufficient information.
If you know two vertices, you can figure out the distance between them, and thereby figure out which of the three sides goes between them. However, the other two sides can be arranged in four different ways.

For example, let's pretend that the "known" side is on the x-axis. (I know, that violates one of your restrictions, but let's pretend, okay?) Then you can have the two remaining sides be:
- above the x-axis with the smaller one to the left
- above the x-axis with the smaller one to the right
- below the x-axis with the smaller one to the left
- below the x-axis with the smaller one to the right.

For your particular problem, if you have specific information given for your starting vertices and side lengths, it's possible that your extra restrictions (no side touches an axis or is parallel/perpendicular to an axis) would eliminate three of the possible configurations, making the fourth one the correct answer. However, you don't provide us with that information, and I really don't feel up to providing a general formula for each possible case.


2.
Also insufficient information.
If you know three vertices, you can hopefully figure out the locations of two of the sides, but even that isn't necessarily the case. (If the distances between AB, BC, and CA all match side lengths of the quadrilateral, things get complicated). Once you know two of the sides, the remaining two sides may either bulge inward or outward (making the quadrilateral concave - V-shaped - or convex. The area should tell you which of these two options is correct. However, if the problems don't provide us with specific numbers for a specific scenario, there are way too many options for me to provide a general formula for each possible case.

I hope you have more specific information that you just didn't provide us, otherwise you're in for a messy task!

Answer 3

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