1. ## 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.

2. 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?

3. Hello, thamathkid1729!

earboth found an elegant solution!

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

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

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

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

... but why?
It was a challenge for me...