Q. 264.5( 2 Votes )

Show that the points A(1,0), B(5,3), C(2,7) and D(-2,4) are the vertices of a rhombus.

Answer :

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


(a) ABCD is a parallelogram, i.e., AC and BD have the same midpoint.


(b) a pair of adjacent edges are equal



Let A(1, 0), B(5, 3), C(2, 7) and D(-2, 4) are the vertices of a rhombus.


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, using Distance Formula


d(A,B)= AB = √(5 – 1)2 + (3 – 0)2


AB = √(4)2 + (3)2


AB = √16 + 9


AB = 25 = 5 units


d(B,C)= BC = √(2 – 5)2 + (7 – 3)2


BC = √(-3)2 + (4)2


BC = √9 + 16


BC = 25 = 5 units


Therefore, adjacent sides are equal.


Hence, ABCD is a rhombus.


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