Пример №13 из задания 20

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)

Пример №13 из задания 20: 31 комментарий

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