Прямоугольник двумя прямолинейными разрезами разбит на четыре малых прямоугольника (см. рис.). Периметры трех из них, начиная с левого верхнего и далее по часовой стрелке, равны \(20\), \(12\) и \(11\). Найдите периметр четвертого прямоугольника.
Решение
Обозначим равные стороны каждого прямоугольника (см. рисунок ниже). Периметр — сумма всех сторон, значит, периметр четвертого прямоугольника будет равен \(P_4=a+d+a+d=2a+2d\).
Распишем, чему равен периметр каждого маленького прямоугольника:
\(P_1=2a+2c=20\);
\(P_2=2b+2c=12\);
\(P_3=2b+2d=11\).
Выразим \(a\) из первого периметра, \(d\) из третьего периметра и подставим в четвертый периметр:
\(2a=20-2c\)
\(2d=11-2b\);
\(P_4=20-2c+11-2b=31-2b-2c\).
Выразим \(b\) из второго периметра и подставим в четвертый:
\(2b=12-2c\);
\(P_4=31-(12-2c)-2c=19\).
Таким, образом получили, что периметр четвертого прямоугольника равен \(19\).
Ответ: \(19\).
Источник: ЕГЭ 2019. Математика. Базовый уровень. Типовые тестовые задания. 14 вариантов заданий. (вариант №2) (Купить книгу)
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