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.

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