Find coordinates for the vertices of alattice rectangle that is three times as long as it is wide, with none of the sides horizontal.
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: $\displaystyle x_1 < x_2,\;y_1 > y_2.)$
$\displaystyle \text{Let: }\,\Delta x =|x_2-x_1|\,\text{ and }\,\Delta y =|y_2-y_1|$Code:| | P | o(x1,y1) | :\ | : \ | ∆y : \ | : \ | : \ | + - - o(x2,y2) | ∆x Q | - - + - - - - - - - - - |
Code:S o * \ * \ * \ * \ * \ * o R * * : P * * :∆x o * - - - -+ :\ * : ∆y : \ * :∆x ∆y : \ *- - - - + : \ * : ∆y : \ * :∆x + - - o - - - -+ ∆x Q ∆y
$\displaystyle \text{We see that }R\text{ is at }\,(x_2 + 3\Delta y,\;y_2 + 3\Delta x)$
. . .$\displaystyle \text{and that }S\text{ is at }\,(x_1 + 3\Delta y,\;y_1+3\Delta x)$