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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