Question

In reperul cartezian xOy se consideră punctele A (1,2) B(-1,1) C (3,5) D(5,a), a apartine lui R. Sa se determine a, stiind ca AB||CD.

Answer 1

$A(1,2);\quad B(-1,1);\quad C(3,5);\quad D(5,a); \\ \\ m_{AB} = \dfrac{y_B-y_A}{x_B-x_A} = \dfrac{1-2}{-1-1} = \dfrac{-1}{-2} = \dfrac{1}{2} \\ \\ m_{CD} = \dfrac{y_D-y_C}{x_D-x_C} = \dfrac{a-5}{5-3} = \dfrac{a-5}{2} \\ \\ AB || CD \overset{(\text{conditia de paralelism dintre doua drepte)}}{\Rightarrow }m_{AB} = m_{CD} \Rightarrow \dfrac{1}{2} = \dfrac{a-5}{2} \Rightarrow \\ \\ \Rightarrow 2 = 2\cdot(a-5) \Rightarrow \\ \\ \Rightarrow a-5 = \dfrac{2}{2} \Rightarrow a-5 = 1 \Rightarrow a = 1+5 \Rightarrow \boxed{a=6}$