I have a rectangle with it's 4 bounds/coordinates on a Cartesian plane. Lets say I want to make the rectangle twice as big. How do i find the new coordinates?
Example, Dotted lines are the new lines after growing the area
In the example image above, the original coordinates are (5, 10) (6, 9) (5, 8) (4, 9). If i wanted to grow the rectangle an arbitrary number like 1.2 times, the new coordinates would be (5, 11) (7, 9) (5, 7) (3, 9). How do i find these new coordinates mathematically?
To double the area, multiply each of the four bounds by $\sqrt{2}$. That will multiply both side lengths by $\sqrt{2}$, and multiply the area by $\sqrt{2}\sqrt{2}=2$.
Example: $x=2, x=4$ and $y=5, y=10$. Current area = $(4-2)\times (10-5)=2\times 5=10$.
Multiply everything by $\sqrt{2}$. The new values will be $x=2\sqrt{2}, x=4\sqrt{2}$ and $y=5\sqrt{2}, y=10\sqrt{2}$. The new area will be $(4\sqrt{2}-2\sqrt{2})\times (10\sqrt{2}-5\sqrt{2})=(2\sqrt{2})\times (5\sqrt{2})=10\sqrt{2}\sqrt{2}=20$.
Second example: $(5, 10) (6, 9) (5, 8) (4, 9) \to (5\sqrt{2}, 10\sqrt{2}) (6\sqrt{2}, 9\sqrt{2}) (5\sqrt{2}, 8\sqrt{2}) (4\sqrt{2}, 9\sqrt{2})$
All four side lengths (in this example not horizontal/vertical) will be $\sqrt{2}$ longer, so the area will be twice as big.