Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite points

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June 21st 2012, 01:50 PM
x3bnm

Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite points

Suppose I'm given $2D$ coordinates of two opposite vertices $B$ and $D$ of an $ABCD$ rectangle. And I know the coordinate of the intersection point $O$ of the diagonals $AC$ and $BD$ of that rectangle.

Is it possible to find the area of the rectangle using this information? If yes what is the procedure for that?
June 21st 2012, 02:12 PM
richard1234

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

No. It's like saying, you're given the length of the hypotenuse of a right triangle. You cannot determine the legs of the triangle given only the hypotenuse.
June 21st 2012, 02:25 PM
x3bnm

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

Thanks richard1234. That's all I wanted to know. I've to find another way to compute the area. Again thanks.
June 21st 2012, 05:39 PM
Soroban

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

Hello, x3bnm!

Quote:

Suppose I'm given coordinates of two opposite vertices $B$ and $D$ of an $ABCD$ rectangle.
And I know the coordinate of the intersection point $O$ of the diagonals $AC$ and $BD$ of that rectangle.

Is it possible to find the area of the rectangle using this information?
If yes, what is the procedure for that?

With no more information, richard1234 is correct.

Code:

* * * * * A * o * * * * B o - - - - * - - - - o D * O * * * o * C * * * * *

Let $BD$ be the diameter of a circle.
Then vertex $A$ can be any point on one semicircle.
(And $C$ is diametrically opposite $A.$ )

If the sides of the rectangle are parallel to the coordinate axes,
. . then a unique solution is possible.
June 22nd 2012, 07:23 PM
x3bnm

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

Quote:

Originally Posted by Soroban

Hello, x3bnm!

With no more information, richard1234 is correct.

Code:

* * * * * A * o * * * * B o - - - - * - - - - o D * O * * * o * C * * * * *

Let $BD$ be the diameter of a circle.
Then vertex $A$ can be any point on one semicircle.
(And $C$ is diametrically opposite $A.$ )

If the sides of the rectangle are parallel to the coordinate axes,
. . then a unique solution is possible.

Yes the rectangle I'm talking about has sides parallel to $X$ and $Y$ axis.

In that case how can I calculate the area of the rectangle if I'm given only coordinates of 1 pair of opposite vertices of the rectangle and the centroid?
June 22nd 2012, 08:31 PM
richard1234

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

If you know that the opposite vertices are (a,b) and (c,d), and that the sides are parallel to the x- and y-axes. Draw the rectangle first. What are the lengths of the base and height?
June 23rd 2012, 08:52 AM
Soroban

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

Hello, x3bnm!

Quote:

$\text{Yes, the rectangle I'm talking about has sides parallel to }x\text{- and }y\text{-axes.}$

Did you take richard1234's suggestion?

Suppose the coordinate are: . $B(p,q),\;D(r,s)$

Plot the two vertices.

Code:

| B | * | (p,q) | | D | * | (r,s) | - - + - - - - - - - - - - - - |

You can see the rectangle, can't you?

Code:

| | (p,q) q * B * - - - - - * C | : : | : : | : : s * A * - - - - - * D | (r,s) | - - + - - - * - - - - - * - - | p r

And you can see that the width is $r-p$ and the height is $q-s.$
June 23rd 2012, 08:52 AM
x3bnm

Re: Find the area of a rectangle:Given coordinates of centroid and 1 pair opposite po

I found the solution to my problem. For those who are interested read on:

Suppose there is a $ABCD$ rectangle(sides are parallel to $X$ and $Y$ axis) where the coordinates of opposite vertices $B$ is $(1,1)$ and the coordinates of $D$ is $(4,3)$ .

We can easily get the coordinates of $A$ as $(1,3)$ and $B$ as $(4,1)$ .

Now it will be easy to calculate the area of rectangle because you have the coordinates of all $4$ vertices.

Thank you richard1234 and Soroban for your help. I really appreciate your help.