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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