The kangaroo is a wild animal which has large, powerful hind legs, a long muscular tail and a small head. There are known over 60 different species of kangaroos and close relatives. Some of them can be two meters tall with weight up to 90 kg. The others are much smaller in size. Kangaroos are unusual and wonderful animals. Kangaroos hop like rabbits to move around quickly. They can jump as far as about three times their own height. They walk on four legs when there is no need to move fast. Kangaroos are called marsupials because they carry their babies in a special bag on their paunches. Kangaroos are herbivore animals, they eat mainly grass and shrubs. They are common in Australia. The kangaroo is one of the most well-known native Australian animals. Although it can also be found in New Guinea kangaroo is the iconic animal of Australia. It is even pictured on the coat of arms for Australia.
Даны координаты вершин треугольника АВС: А(0;-10),В(-12;-1),С(4;12).Найти:
1. Длину стороны АВ:
АВ (с) = √((Хв-Ха)²+(Ув-Уа)²) = √225 = 15.
2. Уравнение сторон АВ и АС:
АВ : Х-Ха = У-Уа х = у + 10
Хв-Ха Ув-Уа -12 9
9х = -12у -120 сократим на 3 и перенесём налево:
3х + 4у + 40 = 0.
у(АВ) = -0,75х - 10.
АС : Х-Ха = У-Уа
Хс-Ха Ус-Уа
11Х - 2У - 20 = 0
у = 5,5х - 10
3. Величину угла А:
cos A= АВ²+АС²-ВС² = 0,4472136.
2*АВ*АС
A = 1,107149 радиан.
A = 63,434949 градусов.
4. Уравнение высоты CD и ее длину.
к(СД) = -1/к(АВ) = -1/(-0,75) = 4/3.
у = (4/3)х + в. Для определения "в" подставим координаты точки С:
12 = (4/3)*4 + в, в = 12 - (16/3) = 20/3.
Уравнение СД: у = (4/3)х + (20/3).
Длину СД можно определить двумя СД = 2S/AB и по координатам точек С и Д.
Приравниваем уравнения АВ и СД: -0,75х - 10 = (4/3)х + (20/3),
(-25/12)х = (20/3) + 10 = 50/3,
х = (50/3)/(-25/12) = (-600/75) = -8,
у = (-3/4)*(-8) - 10 = 6 - 10 = -4. Точка D: (-8; -4).
Длина СД = √((-8-4)² + (-4-12)²) = √(144 + 256) = √400 = 20.
5. Уравнение медианы ВЕ.
Точка Е как середина АС: (2; 1).
ВЕ: Х-Хв = У-Ув х + 12 = у + 1
Хе-Хв Уе-Ув 14 2
Знаменатели сократим на 2: х + 12 = 7у + 7.
Общее уравнение ВЕ: х - 7у + 5 = 0,
с угловым коэффициентом: у = (1/7)х + (5/7).
6. Координаты точки К пересечения медианы ВЕ и высоты CD.
(1/7)х + (5/7) = (4/3)х + (20/3),
(-25/21)х = (125/21)
х = -125/25 = -5, у = (1/7)*(-5) + (5/7) = 0. Точка К:(-5; 0).
7. Уравнение прямой КР, проходящей через точку К параллельно стороне АВ.
Угловой коэффициент равен -0,75.
Уравнение КР: у = (-0,75)х + в. Подставим координаты точки К(-5; 0):
0 = (-0,75)*(-5) + в, в = - (15/4) = -3,75.
у = (-0,75)х - 3,75.
8. Координаты точки М, расположенной симметрично точке А относительно прямой CD.
Так как CD - перпендикуляр к прямой АВ, то точка D(-8; -4) - это та точка, относительно которой требуется найти точку, симметричной точке А.
xM = 2xD - xA = 2*(-8) - 0 = -16,
yM =2yD - yA = 2*(-4) - (-10) = -8 + 10 = 2.
Точка М(-16, 2).
The kangaroo is a wild animal which has large, powerful hind legs, a long muscular tail and a small head. There are known over 60 different species of kangaroos and close relatives. Some of them can be two meters tall with weight up to 90 kg. The others are much smaller in size. Kangaroos are unusual and wonderful animals. Kangaroos hop like rabbits to move around quickly. They can jump as far as about three times their own height. They walk on four legs when there is no need to move fast. Kangaroos are called marsupials because they carry their babies in a special bag on their paunches. Kangaroos are herbivore animals, they eat mainly grass and shrubs. They are common in Australia. The kangaroo is one of the most well-known native Australian animals. Although it can also be found in New Guinea kangaroo is the iconic animal of Australia. It is even pictured on the coat of arms for Australia.