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Консультация № 109354
14.11.2007, 11:14
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Аналитическая геометрия:
Даны вершины треугольника A(4;1); B(7;5); C(-1;1). Составить уравнения биссектрисы внутреннего угла "B" треугольника.
Даны вершины треугольника A(4;1); B(7;5); C(-1;1). Составить уравнения биссектрисы внутреннего угла "B" треугольника.
Обсуждение
Неизвестный
15.11.2007, 04:35
общий
это ответ
Здравствуйте, Petrosoft!
Координаты вектора ВА(-3;-4), |BA| = √(3²+4²) = √25 = 5
координаты вектора ВС(-8;-4), |BC| = √(8²+4²) = √80 = 4√5
Пусть координаты точки D, лежащей на биссектрисе D(x,y) => координаты вектора BD(x-7;y-5)
Угол между двумя векторами = скалярному произведению этих векторов, поделенному на их длины => Угол АBD = BA*BD/(|BA|*|BD|); угол СBD = BС*BD/(|BС|*|BD|)
Так как биссектриса делит углы пополам => угол CBD = углу ABD => BA*BD/(|BA|*|BD|) = BС*BD/(|BС|*|BD|) => BA*BD/|BA| = BС*BD/|BС|
=> (-3;-4)*(x-7;y-5)/5 = (-8;-4)*(x-7;y-5)/√80
(-3x+21-4y+20)/5 = (-8x+56-4y+20)√5/20
-3x-4y+41 = (-2x-y+19)√5
(3-2√5)x + (4-√5)y + 19√5-41 = 0
(2-5√5)x +11y + 35√5-69 = 0 - уравнение биссектрисы
Координаты вектора ВА(-3;-4), |BA| = √(3²+4²) = √25 = 5
координаты вектора ВС(-8;-4), |BC| = √(8²+4²) = √80 = 4√5
Пусть координаты точки D, лежащей на биссектрисе D(x,y) => координаты вектора BD(x-7;y-5)
Угол между двумя векторами = скалярному произведению этих векторов, поделенному на их длины => Угол АBD = BA*BD/(|BA|*|BD|); угол СBD = BС*BD/(|BС|*|BD|)
Так как биссектриса делит углы пополам => угол CBD = углу ABD => BA*BD/(|BA|*|BD|) = BС*BD/(|BС|*|BD|) => BA*BD/|BA| = BС*BD/|BС|
=> (-3;-4)*(x-7;y-5)/5 = (-8;-4)*(x-7;y-5)/√80
(-3x+21-4y+20)/5 = (-8x+56-4y+20)√5/20
-3x-4y+41 = (-2x-y+19)√5
(3-2√5)x + (4-√5)y + 19√5-41 = 0
(2-5√5)x +11y + 35√5-69 = 0 - уравнение биссектрисы
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