Question

Determinati coordonatele varfului D al paralelogramului ABCD stiind ca A(-2,-1),B(3,2) si C(5,1)

Answer 1

$A(-2,-1); B(3,2); C(5,1); D(a,b) \\ \\ Daca reprezentam cele 3 puncte in sistemul xOy observam ca \overrightarrow{AB} = \overrightarrow{DC} \\ \\ \Rightarrow (x_B-x_A)i + (y_B-y_A)j = (x_C - x_D)i+(y_C-y_D)j \Rightarrow \\ \Rightarrow \big(3-(-2)\big)i+ \big(2-(-1)\big)j = (5-a)i + (1-b)j \Rightarrow \\ \Rightarrow (3+2)i+(2+1)j = (5-a)i + (1-b)j \Rightarrow \\ \Rightarrow 5i+3j = (5-a)i + (1-b)j\Rightarrow$

$\Rightarrow \left\{ \begin{array}{c} 5-a = 5 \\1-b = 3 \end{array} \right \Rightarrow \ \left\{ \begin{array}{c} a = 5-5 \\b = 1-3 \end{array} \right \Rightarrow \left\{ \begin{array}{c} a = 0 \\b = -2 \end{array} \right | \Rightarrow \boxed{D(0,-2)}$