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Консультация № 150306
12.11.2008, 12:11
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Координаты треугольника на плоскости: А(5,2) В (4,-1) С (-4,1)
Найти:
1. Длину и уравнение высоты, опущенной из вершины А
2. Систему неравенств, задающих внутренность треугольника АВС
Найти:
1. Длину и уравнение высоты, опущенной из вершины А
2. Систему неравенств, задающих внутренность треугольника АВС
Обсуждение
Неизвестный
12.11.2008, 15:30
общий
это ответ
Здравствуйте, Гудилина Инга!
Напишем уравнение стороны ВС:
(x-4)/(-4-4)=(y+1)/(1+1)
(x-4)/(-8)=(y+1)/2
x-4=-4y-4
x+4y=0
Нормальный вектор этой прямой с координатами (1, 4) будет направляющим вектором искомой высоты. Поэтому можем написать уравнение высоты через точку А(5, 2) и направляющий вектор:
(x-5)/1=(y-2)/4
4x-20=y-2
4x-y-18=0
Длина высоты:
d=|1*5+4*2|/sqrt(1^2+4^2)=13/sqrt17
Или найдем точку пересечения высоты с ВС - т.Н
x=-4y
4(-4y)-y-18=0
-17y=18
y=-18/17
x=72/17
H(72/17, -18/17)
А теперь найдем расстояние между точками:
AH=sqrt((72/17-5)^2+(-18/17-2)^2)=sqrt(169/289+2704/289)=sqrt(2873/289)=sqrt(169/17)=13/sqrt17
2. AB:
(x-5)/(4-5)=(y-2)/(-1-2)
-3x+15=-y+2
y=3x-13
AC:
(x-5)/(-4-5)=(y-2)/(1-2)
-x+5=-9y+18
9y=x+13
y=1/9*(x+13)
BC:
x+4y=0
y=-1/4*x
Система неравенств, задающих внутренность треугольника (включая и сами стороны треугольника)
y>=3x-13
y>=-1/4*x
y<=1/9*(x+13)
Напишем уравнение стороны ВС:
(x-4)/(-4-4)=(y+1)/(1+1)
(x-4)/(-8)=(y+1)/2
x-4=-4y-4
x+4y=0
Нормальный вектор этой прямой с координатами (1, 4) будет направляющим вектором искомой высоты. Поэтому можем написать уравнение высоты через точку А(5, 2) и направляющий вектор:
(x-5)/1=(y-2)/4
4x-20=y-2
4x-y-18=0
Длина высоты:
d=|1*5+4*2|/sqrt(1^2+4^2)=13/sqrt17
Или найдем точку пересечения высоты с ВС - т.Н
x=-4y
4(-4y)-y-18=0
-17y=18
y=-18/17
x=72/17
H(72/17, -18/17)
А теперь найдем расстояние между точками:
AH=sqrt((72/17-5)^2+(-18/17-2)^2)=sqrt(169/289+2704/289)=sqrt(2873/289)=sqrt(169/17)=13/sqrt17
2. AB:
(x-5)/(4-5)=(y-2)/(-1-2)
-3x+15=-y+2
y=3x-13
AC:
(x-5)/(-4-5)=(y-2)/(1-2)
-x+5=-9y+18
9y=x+13
y=1/9*(x+13)
BC:
x+4y=0
y=-1/4*x
Система неравенств, задающих внутренность треугольника (включая и сами стороны треугольника)
y>=3x-13
y>=-1/4*x
y<=1/9*(x+13)
Форма ответа
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