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Консультация № 197440
15.12.2019, 13:14
0.00 руб.
16.12.2019, 10:07
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1
Здравствуйте! Прошу помощи в следующем вопросе:
В треугольнике ABC известно, что AB = 11, AC = 23, а медиана AM = 10. Найдите сторону BC и угол BAC треугольника
В треугольнике ABC известно, что AB = 11, AC = 23, а медиана AM = 10. Найдите сторону BC и угол BAC треугольника
Обсуждение
17.12.2019, 12:13
общий
это ответ
Здравствуйте, pilot!
Дано: AB=11 , AC=23, медиана AM = 10.
Вычислить сторону BC и угол BAC .
Решение : Листаем тему "Треугольники" в школьном учебнике или справочнике по математике. Находим свойства медиан, цитирую: "Медианы, проведенные из вершин A, B, C, обозначаются соответственно: ma, mb, mc .
mc = [$8730$](2a2 + 2b2 - c2) / 2"
Корректируем эту обобщённую формулу применительно к заданной в Условии медиане AM = ma :
ma = [$8730$](2b2 + 2c2 - a2) / 2
Здесь ma = 10, b = AC = 23 , c = AB = 11 .
Решаем квадратное уравнение и получаем искомую сторону a=BC=30 . Второй, отрицательный корень игнорируем, как неподходящий для реального треугольника.
Теорему косинусов c2 = a2 + b2 - 2·a·b·cos(C) переводим на имена наших переменных :
2b·c·cos(BAC) + a2 = b2 + c2
вычисляем косинус угла BAC : CosC = (b2 + c2 - a2) / (2·b·c) = -0,494 , что соответствует углу BAC в 120 градусов.
Ответ : сторона BC равна 30 (единиц), угол BAC равен 120°.
Для проверки правильности решения полезно построить треугольник. Вычисления и графики удобно делать в бесплатном приложении ru.wikipedia.org/wiki/Mathcad . Маткад-скриншот прилагаю. Я добавил в него подробные комментарии зелёным цветом.
Дано: AB=11 , AC=23, медиана AM = 10.
Вычислить сторону BC и угол BAC .
Решение : Листаем тему "Треугольники" в школьном учебнике или справочнике по математике. Находим свойства медиан, цитирую: "Медианы, проведенные из вершин A, B, C, обозначаются соответственно: ma, mb, mc .
mc = [$8730$](2a2 + 2b2 - c2) / 2"
Корректируем эту обобщённую формулу применительно к заданной в Условии медиане AM = ma :
ma = [$8730$](2b2 + 2c2 - a2) / 2
Здесь ma = 10, b = AC = 23 , c = AB = 11 .
Решаем квадратное уравнение и получаем искомую сторону a=BC=30 . Второй, отрицательный корень игнорируем, как неподходящий для реального треугольника.
Теорему косинусов c2 = a2 + b2 - 2·a·b·cos(C) переводим на имена наших переменных :
2b·c·cos(BAC) + a2 = b2 + c2
вычисляем косинус угла BAC : CosC = (b2 + c2 - a2) / (2·b·c) = -0,494 , что соответствует углу BAC в 120 градусов.
Ответ : сторона BC равна 30 (единиц), угол BAC равен 120°.
Для проверки правильности решения полезно построить треугольник. Вычисления и графики удобно делать в бесплатном приложении ru.wikipedia.org/wiki/Mathcad . Маткад-скриншот прилагаю. Я добавил в него подробные комментарии зелёным цветом.
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