$h^2 + \left( \dfrac{T - a}{2} \right)^2 = \left( \dfrac{L - a}{2} \right)^2$
$h = \sqrt{\dfrac{(L - a)^2}{4} - \dfrac{(T - a)^2}{4}}$
$h = \frac{1}{2} \sqrt{(L - a)^2 - (T - a)^2}$
$h = \frac{1}{2} \sqrt{L^2 - 2La + a^2 - T^2 + 2Ta - a^2}$
$h = \frac{1}{2} \sqrt{L^2 - T^2 - 2(L - T)a}$
Area:
$A = \frac{1}{2}(a + T)h$
$A = \frac{1}{2}(a + T)\left( \frac{1}{2} \sqrt{L^2 - T^2 - 2(L - T)a} \right)$
$A = \frac{1}{4}(a + T) \sqrt{L^2 - T^2 - 2(L - T)a}$ (note that L and T are constant)
$\dfrac{dA}{da} = \dfrac{1}{4} \left[ (a + T) \dfrac{-2(L - T)}{2\sqrt{L^2 - T^2 - 2(L - T)a}} + \sqrt{L^2 - T^2 - 2(L - T)a} \right] = 0$
$\sqrt{L^2 - T^2 - 2(L - T)a} = \dfrac{(a + T)(L - T)}{\sqrt{L^2 - T^2 - 2(L - T)a}}$
$L^2 - T^2 - 2(L - T)a = (a +T)(L - T)$
$L^2 - T^2 - 2La + 2Ta = La - Ta + TL - T^2$
$L^2 - TL = 3La - 3Ta$
$3(L - T)a = L(L - T)$
$a = \frac{1}{3}L$
Base = 1/3 × length of strip answer