ВН – высота. \(BC = 7,\,\,\,\,\,AD = 13,\,\,\,\,\,\,{S_{ABCD}} = 40.\)
\({S_{ABCD}} = \frac{{BC + AD}}{2} \cdot BH\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,40 = \frac{{7 + 13}}{2} \cdot BH\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,BH = 4.\)
\(AH = \frac{{AD — BC}}{2} = \frac{{13 — 7}}{2} = 3.\)
По теореме Пифагора из треугольника АВН:
\(A{B^2} = A{H^2} + B{H^2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,A{B^2} = {3^2} + {4^2} = 25\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,AB = 5.\)
Тогда: \({p_{ABCD}} = AB + BC + CD + AD = 5 + 7 + 5 + 13 = 30.\)
Ответ: 30.