# Coordinate Geometry

• Mar 7th 2011, 02:19 PM
thamathkid1729
Coordinate Geometry
Find coordinates for the vertices of a
lattice rectangle that is three times as long as it is wide, with none of the sides horizontal.

• Mar 8th 2011, 12:12 AM
earboth
Quote:

Originally Posted by thamathkid1729
Find coordinates for the vertices of a
lattice rectangle that is three times as long as it is wide, with none of the sides horizontal.

A(0, 0)
B(12, 9)
C(15, 5)
D(3, -4)

... but why?
• Mar 8th 2011, 05:10 AM
Soroban
Hello, thamathkid1729!

earboth found an elegant solution!

Quote:

Find coordinates for the vertices of a lattice rectangle
that is three times as long as it is wide, with none of the sides horizontal.

Select any two points: $\displaystyle P(x_1,y_1),\;Q(x_2,y_2)\,\text{ with }\,x_1 \ne x_2,\;y_1 \ne y_2$
. . (In this example: $\displaystyle x_1 < x_2,\;y_1 > y_2.)$

Code:

       |       |    P       |    o(x1,y1)       |    :\       |    : \       |  ∆y :  \       |    :  \       |    :    \       |    + - - o(x2,y2)       |      ∆x  Q       |   - - + - - - - - - - - -       |
$\displaystyle \text{Let: }\,\Delta x =|x_2-x_1|\,\text{ and }\,\Delta y =|y_2-y_1|$

Code:

                                      S                                     o                                 *  \                             *        \                           *            \                        *                \                     *                    \                 *                      o R               *                      *  :       P  *                      *    :∆x         o                      * - - - -+         :\                  *  :  ∆y         : \              *    :∆x     ∆y :  \          *- - - - +         :  \      *  :    ∆y         :    \  *    :∆x         + - - o - - - -+           ∆x  Q    ∆y

$\displaystyle \text{We see that }R\text{ is at }\,(x_2 + 3\Delta y,\;y_2 + 3\Delta x)$

. . .$\displaystyle \text{and that }S\text{ is at }\,(x_1 + 3\Delta y,\;y_1+3\Delta x)$

• Mar 8th 2011, 02:08 PM
thamathkid1729
Quote:

Originally Posted by earboth
A(0, 0)
B(12, 9)
C(15, 5)
D(3, -4)

... but why?

It was a challenge for me...(Blush)