Prove that the point (4,8), (0,2), (3,0) and (7,6) are the vertices of a rectangle.


Note that to show that a quadrilateral is a rectangle, it is sufficient to show that


(a) ABCD is a parallelogram, i.e., AC and BD bisect each other and,


(b) the diagonal AC and BD are equal



Let A(4, 8), B(0, 2), C(3, 0) and D(7, 6) are the vertices of a rectangle.


Coordinates of the midpoint of AC are



Coordinates of the midpoint of BD are



Thus, AC and BD have the same midpoint.


Hence, ABCD is a parallelogram


Now, check for the diagonals by using the distance formula


AC = √(3 – 4)2 + (0 – 8)2


= √(-1)2 + (-8)2


= √1 + 64


= √65 units


and


BD = √(7 - 0)2 + (6 – 2)2


⇒ BD = √(7)2 + (4)2


⇒ BD = √49 + 16


⇒ BD = √65 units


∴ AC = BD


Hence, ABCD is a rectangle.


27
1