Question: Prove that the mid points of the sides of a rectangle form a rhombus.



Steps to derive

1 The rectangle is as shown.


2 Coordinates of M = (a+c2, b)

3 Coordinates of N = (c, b+d2)

4 Coordinates of O = (a+c2, d)

5 Coordinates of P = (a, b+d2)

6 MN = (a+c2-c)2+(b -b+d2)2
= (a-c2)2+(b-d2)2

7 OP = (a+c2-a)2+(d -b+d2)2
= (c-a2)2+(d-b2)2 ⇒ (a-c2)2+(b-d2)2

8 MN = OP

9 ON = (a+c2-c)2+(d -b+d2)2
= (a-c2)2+(d-b2)2

10 MP = (a+c2-a)2+(b -b+d2)2
= (c-a2)2+(b-d2)2 ⇒ (a-c2)2+(d-b2)2

11 ON = PM

12 MN = OP = ON = PM.

13 All the sides are equal.

14 MNOP is a rhombus.

15 Mid points of teh rectangle form a rhombus.



Hence the right answer is Option

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