Question

verifique se os pontos indicados em cada item são colineares
a) A(-12,-3), B(-2,3) e C(3,6)

b) (-2,-2), B(0,4) e C(1,7)

Answer 1

Para verificar a colinearidade de 3 pontos, podemos utilizar o determinante de uma matriz 3x3. Caso o determinante resulte em 0 (zero) os pontos estarão alinhados.

$\left|\begin{array}{ccc}x_A&y_A&1\\x_B&y_B&1\\x_C&y_C&1\end{array}\right|~=~0$

a) São colineares

$\left|\begin{array}{ccc}-12&-3&1\\-2&3&1\\3&6&1\end{array}\right| =\\\\\\=~(-12~.~3~.~1+(-2)~.~6~.~1+3~.~(-3)~.~1)-(1~.~3~.~3+1~.~6~.~(-12)+1~.~(-3)~.~(-2))\\\\\\=~(-36-12-9)-(9-72+6)\\\\\\=~(-57)-(-57)\\\\\\=~-57+57\\\\\\=~0$

b)São colineares

$\left|\begin{array}{ccc}-2&-2&1\\0&4&1\\1&7&1\end{array}\right| =\\\\\\=~(-2~.~4~.~1+0~.~7~.~1+1~.~(-2)~.~1)-(1~.~4~.~1+1~.~7~.~(-2)+1~.~(-2)~.~0)\\\\\\=~(-8+0-2)-(4-14+0)\\\\\\=~(-10)-(-10)\\\\\\=~-10+10\\\\\\=~0$