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

• Jun 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 $\displaystyle 2D$ coordinates of two opposite vertices $\displaystyle B$ and $\displaystyle D$ of an $\displaystyle ABCD$ rectangle. And I know the coordinate of the intersection point $\displaystyle O$ of the diagonals$\displaystyle AC$ and $\displaystyle BD$ of that rectangle.

Is it possible to find the area of the rectangle using this information? If yes what is the procedure for that?
• Jun 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.
• Jun 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.
• Jun 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 $\displaystyle B$ and $\displaystyle D$ of an $\displaystyle ABCD$ rectangle.
And I know the coordinate of the intersection point $\displaystyle O$ of the diagonals$\displaystyle AC$ and $\displaystyle BD$ of that rectangle.

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

Code:

              * * *           *          *  A         *              o       *                *       *                  *     B o - - - - * - - - - o D       *        O        *       *                *         o                *       C  *          *               * * *
Let $\displaystyle BD$ be the diameter of a circle.
Then vertex $\displaystyle A$ can be any point on one semicircle.
(And $\displaystyle C$ is diametrically opposite $\displaystyle A.$)

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

Code:

              * * *           *          *  A         *              o       *                *       *                  *     B o - - - - * - - - - o D       *        O        *       *                *         o                *       C  *          *               * * *
Let $\displaystyle BD$ be the diameter of a circle.
Then vertex $\displaystyle A$ can be any point on one semicircle.
(And $\displaystyle C$ is diametrically opposite $\displaystyle A.$)

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 $\displaystyle X$ and $\displaystyle Y$ 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?
• Jun 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?
• Jun 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:

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

Did you take richard1234's suggestion?

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

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 $\displaystyle r-p$ and the height is $\displaystyle q-s.$

• Jun 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 $\displaystyle ABCD$ rectangle(sides are parallel to $\displaystyle X$ and $\displaystyle Y$ axis) where the coordinates of opposite vertices $\displaystyle B$ is $\displaystyle (1,1)$ and the coordinates of $\displaystyle D$ is $\displaystyle (4,3)$.

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

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

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