1. ## circles

Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance between the centers of the two circles?

2. Originally Posted by sri340
Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance between the centers of the two circles?
Let $\displaystyle P=(x_1,y_1)$ be the center of one circle, while $\displaystyle Q=(x_2,y_2)$ is the center of the other. It is clear that the distance between $\displaystyle P$ and $\displaystyle Q$ willl be greatest when the circles lay at opposite corners of the rectangle. On a coordinate grid, locate $\displaystyle P$ and $\displaystyle Q$ and then determine the distance $\displaystyle d$ by

$\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

3. Or...

The circles touch opposite ends of the rectangle, so their centres are 6 inches from either end.
6+6=12 inches from side to centre combined when the centres are furthest apart. This leaves at most 2 inches from centre to centre.
Therefore the maximum distance between the centres is 2 inches.

4. Originally Posted by Archie Meade
Or...

The circles touch opposite ends of the rectangle, so their centres are 6 inches from either end.
6+6=12 inches from side to centre combined when the centres are furthest apart. This leaves at most 2 inches from centre to centre.
Therefore the maximum distance between the centres is 2 inches.
No.

5. Apologies,
i used a radius of 6.
Pythagoras' theorem gives $\displaystyle \sqrt{(14-6)^2+(12-6)^2}$

6. Originally Posted by Archie Meade
Apologies,
i used a radius of 6.
Pythagoras' theorem gives $\displaystyle \sqrt{(14-6)^2+(12-6)^2}$
That's okay, but I don't follow. So, you are using $\displaystyle (14,12)$ as one of your points?

Even if you used the correct radius this would yield an incorrect result. Let's wait and see if the OP is going to reply. If you like, we can talk about this via IM.

EDIT:Oh wait. I see...it works.

7. Hello, sri340!

Did you make a sketch?
You can almost "eyeball" the answer!

Two circles are drawn in a 12-inch by 14-inch rectangle.
Each circle has a diameter of 6 inches.
If the circles do not extend beyond the rectangular region,
what is the greatest possible distance between the centers of the two circles?
Code:
      : - 3 - : - - - -8- - - - : - 3 - :
- *-------o-------------------------* -
: |   o   :   o                     | :
: |o      :3     o                  | 3
: |      P:                 R       | :
: o - - - o - - - o - - - - *       | -
: |   3     *               :       | :
: |o          *  o          :       | :
: |   o       o *           :       | :
12 |       o       *         :       | 6
: |                 *       o       | :
: |                   * o   :   o   | :
: |                  o  *   :      o| :
: |                       * :Q      | :
: |                 o       o - - - o -
: |                         :   3   | :
: |                  o     3:      o| 3
: |                     o   :   o   | :
- *-------------------------*-------* -
: - - - - - - - 14  - - - - - - - :

The circles have centers $\displaystyle P$ and $\displaystyle Q$ and radius 3.
They are placed in opposite corners of the rectangle.

Center P and Q with point R form a right triangle
. . with legs: .$\displaystyle PR = 8,\:QR = 6$

Therefore: .$\displaystyle PQ \,=\,10$