Question

Points A(-2, 4), B(1, 3), C(4, -1) and D form a parallelogram. What are the coordinates of D?

Answer 1

Answer:

$\text{The fourth vertex is (1,0)}$

Step-by-step explanation:

$\text{Given points are A(-2, 4), B(1, 3), C(4, -1)}$

$\text{Let the fourth vertex D be (x,y)}$

$\boxed{\begin{minipage}{8cm}We know that diagonals of a parallelogram bisect each other\end{minipage}}$

$\implies\,\text{Midpoint of diagonal AC }=\text{Midpoint of diagonal BD }$

Using midpoint formula,

$\textbf{The midpoint of the line joining }(x_1,y_1)\text{ and }(x_2,y_2)\textbf{ is }\bf(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$\implies\,(\frac{-2+4}{2},\frac{4-1}{2})=(\frac{1+x}{2},\frac{3+y}{2})$

$\implies\,(\frac{2}{2},\frac{4-1}{2})=(\frac{1+x}{2},\frac{3+y}{2})$

$\implies\,(1,\frac{3}{2})=(\frac{1+x}{2},\frac{3+y}{2})$

$\text{Equating the corresponding coordinates}$

$1=\frac{1+x}{2}$

$\implies\,2=1+x$

$\implies\,x=1$

$\implies\,\frac{3}{2}=\frac{3+y}{2}$

$\implies\,3=3+y$

$\implies\,y=0$

$\therefore\text{The fourth vertex is (1,0)}$