11.32. The area of a trapezoid is 1. What is the smallest value that the largest diagonal of this trapezoid can have?
11.32. Let the lengths of the diagonals of the trapezoid be denoted by \(d_{1}\) and \(d_{2}\), the lengths of their projections on the base by \(p_{1}\) and \(p_{2}\), the lengths of the bases by \(a\) and \(b\), and the height by \(h\). Let \(d_{1} \geqslant d_{2}\) for definiteness. Then \(p_{1} \geqslant p_{2}\). It is clear that \(p_{1}+p_{2} \geqslant a+b\). Therefore, \(p_{1} \geqslant (a+b) / 2 = S / h = 1 / h\). Consequently, \(d_{1}^{2} = p_{1}^{2} + h^{2} \geqslant \frac{1}{h^{2}} + h^{2} \geqslant 2\), and equality is achieved only if \(p_{1} = p_{2} = h = 1\). In this case, \(d_{1} = \sqrt{2}\).