Question

The points $(0,0)$ and $(4,4)$ are the extremities of a diagonal of a parallelogram. If the third vertex is $(4,0)$. Find the coordinates of the fourth vertex.

• A. $(0,4)$
• B. $(-3,1)$
• C. $(-2,4)$
• D. None of these

Answer 1

The correct option is A $(0,4)$

Given $A(0,0),B(4,0),C(4,4)$ Let the fourth coordinate is $P(x,y)$.

Since the opposite sides of a parallelogram are equal.

So, $AB=PC$ and $BC=PA$.

So by distance formula we have,

Distance between two points = $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Hence $4=\sqrt{(4-x{)}^2+(4-y{)}^2}$............$\left(1\right)$

and $4=\sqrt{(x{)}^2+(y{)}^2}$.........$\left(2\right)$

Solving $\left(1\right)$ and $\left(2\right)$:

$\Rightarrow x+y=4$.........$\left(3\right)$.

So point $P$ lies on this line at a distance $4$ units from $C$ and $A$.

Hence, the required point is $P(0,4)$.