ВН – высота. \(BC = 12,\,\,\,\,\,\,AD = 27,\,\,\,\,\,\,\angle \,A = {60^ \circ }.\)
Тогда: \(AH = \frac{{AD — BC}}{2} = \frac{{27 — 12}}{2} = \frac{{15}}{2} = 7,5.\)
По определению косинуса из треугольника АВН:
\(\cos A = \frac{{AH}}{{AB}}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\cos {60^ \circ } = \frac{{7,5}}{{AB}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\frac{1}{2} = \frac{{7,5}}{{AB}}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,AB = 15.\)
\({p_{ABCD}} = AB + BC + CD + AD = 15 + 12 + 15 + 27 = 69.\)
Ответ: 69.