Answer: The required co-ordinates of vertex D are (0, 4).
Step-by-step explanation: Given that the co-ordinates of three vertices of a parallelogram ABCD are A(-1,0), B(4,0) and C(5,4).
We are to find the co-ordinates of vertex D.
Let (a, b) be the co-ordinates of the vertex D.
Since the opposite sides of a parallelogram are parallel to each other and the slopes of two parallel lines are equal, so we must have
[tex]\textup{slope of AB}=\textup{slope of CD}\\\\\\\Rightarrow \dfrac{0-0}{4-(-1)}=\dfrac{b-4}{a-5}\\\\\\\Rightarrow \dfrac{0}{5}=\dfrac{b-4}{a-5}\\\\\\\Rightarrow 0=\dfrac{b-4}{a-5}\\\\\Rightarrow b-4=0\\\\\Rightarrow b=4[/tex]
and
[tex]\textup{slope of BC}=\textup{slope of AD}\\\\\\\Rightarrow \dfrac{4-0}{5-4}=\dfrac{b-0}{a-(-1)}\\\\\\\Rightarrow \dfrac{4}{1}=\dfrac{b}{a+1}\\\\\\\Rightarrow 4=\dfrac{4}{a+1}\\\\\\\Rightarrow 1=\dfrac{1}{a+1}\\\\\Rightarrow a+1=1\\\\\Rightarrow a=1-1\\\\\Rightarrow a=0.[/tex]
Thus, the required co-ordinates of vertex D are (0, 4).
