I'm trying to find an expression, equation.
Given rectangle x by y, what is the single value, a, by which I can reduce
each side to halve the area.
xy / 2 = (x-a)(y-a) solve for a
Hello, gowan!
Given rectangle $\displaystyle x$ by $\displaystyle y$, what is the single value, $\displaystyle a$,
by which I can reduce each side to halve the area?
. . $\displaystyle (x-a)(y-a)\:=\:\frac{xy}{2}\qquad \hdots$ Solve for $\displaystyle a.$
We have: .$\displaystyle xy - ax - ay + a^2 \:=\:\frac{xy}{2} \quad\Rightarrow\quad 2xy -2ax - 2ay + 2a^2 \:=\:xy$
. . which simplifies to:. . $\displaystyle 2a^2 - 2(x+y)a + xy \:=\:0$ . . . a quadratic in $\displaystyle a$
Quadratic Formula: .$\displaystyle a \;=\;\frac{2(x+y) \pm\sqrt{4(x+y)^2 - 8xy}}{4} \;=\;\frac{(x+y) \pm\sqrt{x^2+y^2}}{2}$
The answer is the smaller of the two values: .$\displaystyle a \;=\;\frac{(x+y)-\sqrt{x^2+y^2}}{2} $