Wyznacz równania prostych w których zawieraja sie boki trojkata o wierzchołkach A,B,C .Czy jest to t.... Question from @anaaa94

September 2018 1 25 Report

Wyznacz równania prostych w których zawieraja sie boki trojkata o wierzchołkach A,B,C .Czy jest to trójkąt prostokątny ?

A(-6,0) B(1,1) C(2,4)

Proszę o wytłumaczenie i rysunek ;)

bonizaur

Ogólne równanie prostej przechodzącej przez dwa punkty.

$(y-y_A)(x_B-x_A)-(y_B-y_A)(x-x_A)=0$

Dla punktów A i B mamy zatem prostą AB

$(y-0)(1-(-6))-(1-0)(x-(-6))=0\\ 7y-x-6=0\\ 7y=x+6\\ y=\frac{x+6}{7}$

Dla punktów A i C mamy zatem prostą AC

$(y-0)(2-(-6))-(4-0)(x-(-6))=0\\8y-4x-24=0\\8y=4x+24\\y=\frac{x}{2}+3$

Dla punktów B i C mamy zatem prostą BC

$(y-1)(2-1)-(4-1)(x-1)=0\\y-1-3x+3=0\\y-3x+2=0\\y=3x-2$

Z warunku prostopadłości dwóch prostych mamy, że dla równania w postaci

$A_1 x+B_1 y+C_1=0 ,\ \ A_2 x+B_2 y+C_2=0$

są one prostopadłe gdy

$A_1 A_2+B_1 B_2=0$

Sprawdzamy czy proste AB i BC są prostopadłe:

$|AB|\ \ -x+7y-6=0\\ |BC|\ \ -3x+y+2=0\\ (-1)(-3)+(7)(1)=0\\ 3+7\neq0$

Te nie są prostopadłe

$|AB|\ \ -x+7y-6=0\\ |AC|\ \ -4x+8y-24=\\ (-1)(-4)+(7)(8)=0\\ 4+56\neq0$

Te też nie są.

$|BC|\ \ -3x+y+2=0\\ |AC|\ \ -4x+8y-24=\\ (-3)(-4)+(1)(8)=0\\ 12+8\neq0$

Te również.

Zatem trójkąt nie jest prostokątny, gdyż żadna z prostych nie jest prostopadła do innej.

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