Если вершины треугольника заданы, как точки в прямоугольной декартовой системе координат: A1(x1,y1), A2(x2,y2), A3(x3,y3), то площадь такого треугольника можно вычислить по формуле определителя второго порядка:
Abdilda Tazhibayev was born in 1909 in Akmeshit (now Kyzylorda). Medium
one hundred
The Kipchak tribe Tory clan is descended from Bessary's grandfather. [1] [2] His father, Tazhibai, died in 1915 and was left in the care of his mother, Aimankul. Due to his condition, six-year-old Abdilda was taken to a suburb of Karaozek. Then the future poet lived until the age of thirteen with his mother Aimankul and uncle Daldabay
grows in upbringing. Young Abdilda's poems and literature
His mother and aunt also aroused his curiosity
will be the grandfather. Aimankul is fluent in Arabic and Persian
He was literate, open-minded, and cheerful.
He writes letters to Abdilda and teaches her literature. Abdilda's mother Aimankul, who later became a well-known folk poet and published a collection of poems, at the request of his father Daldabay, recited to the villagers the stories of the East and various Kazakh poems. Passionate, silent, silent
One of his listeners was Abdil. Later, in a small group, Abdilda would recite the stories and epics to his guests and villagers.
Даны вершины треугольника A(−2,1), B(3,3), С(1,0). Найти:
а) длина стороны AB = √((3-(-2))² + (3-1)² = √(25 + 4) = √29.
б) уравнение медианы BM.
Находим координаты точки М как середины стороны АС.
М(((-2+1)/2; (1+3)/2) = (-0,5; 2).
Вектор ВМ = ((-0,5-3); (2-3)) = (-3,5; -1).
Уравнение ВМ: (х – 3)/(-3,5) = (у – 3)/(-1). Это в каноническом виде.
Оно же в общем виде 7у – 2х – 15 = 0.
И в виде уравнения с угловым коэффициентом у = (2/7)х + (15/7).
в) cos угла BCA.
Вектор СВ = ((1-3); (0-3)) = (-2; -3). Модуль равен √(4 + 9) = √13.
Вектор СА = ((1-(-2)); (0-1)) = (3; -1). Модуль равен √(9 + 1) = √10.
cos(BCA) = (-2*3 + (-3)*(-1))/( √13*√10) = -3/√130 ≈ -0,26312.
г) уравнение высоты CD.
Находим уравнение стороны АВ.
Вектор AB = ((3-(-2)); (3-1)) = (5; 2).
Уравнение АВ: (х + 2)/5 = (у -1)/2 или у = (2/5)х + (9/5).
Угловой коэффициент перпендикуляра к АВ (это высота СD) равен -1/(2/5) = -5/2. Подставим координаты точки С.
0 = (-5/2)*1 + b. Отсюда b = 5/2.
Уравнение CD: y = (-5/2)x + (5/2).
д) длина высоты СD.
Для вычисления расстояния от точки M(Mx; My) до прямой Ax + By + C = 0 используем формулу:
d = (A·Mx + B·My + C)/√A2 + B2
Подставим в формулу данные: координаты точки С(1; 0) и уравнение прямой АВ:
2х – 5у + 9 = 0.
d = (2·1 + (-5)·0 + 9)/√22 + (-5)2 = (2 + 0 + 9)/√4 + 25 =
= 11/√29 = 11√29/29 ≈ 2.0426487.
е) площадь треугольника АВС по векторам.
Если вершины треугольника заданы, как точки в прямоугольной декартовой системе координат: A1(x1,y1), A2(x2,y2), A3(x3,y3), то площадь такого треугольника можно вычислить по формуле определителя второго порядка:
S= ± (1 /2) *(x1−x3 y1−y3 )
(x2−x3 y2−y3 )
x1−x3 y1−y3
x2−x3 y2−y3
A(−2,1), B(3,3), С(1,0).
S = (1/2)}|((-2-1)*(3-0) – (1-0)*3-1))| = (1/2)*|(-9-2)| = 11/2 = 5,5 кв.ед.
Abdilda Tazhibayev was born in 1909 in Akmeshit (now Kyzylorda). Medium
one hundred
The Kipchak tribe Tory clan is descended from Bessary's grandfather. [1] [2] His father, Tazhibai, died in 1915 and was left in the care of his mother, Aimankul. Due to his condition, six-year-old Abdilda was taken to a suburb of Karaozek. Then the future poet lived until the age of thirteen with his mother Aimankul and uncle Daldabay
grows in upbringing. Young Abdilda's poems and literature
His mother and aunt also aroused his curiosity
will be the grandfather. Aimankul is fluent in Arabic and Persian
He was literate, open-minded, and cheerful.
He writes letters to Abdilda and teaches her literature. Abdilda's mother Aimankul, who later became a well-known folk poet and published a collection of poems, at the request of his father Daldabay, recited to the villagers the stories of the East and various Kazakh poems. Passionate, silent, silent
One of his listeners was Abdil. Later, in a small group, Abdilda would recite the stories and epics to his guests and villagers.