Question

In the figure, the midpoints of the diagonals of a rhombus are joined to form a small quadrilateral:

i. Prove that this quadrilateral is a rhombus.

ii. The area of the small rhombus is 3 square centimetres. What is the area of the large rhombus?

Answer 1

The diagonals of the rhombus intersect at O and they bisect each other at right angles. OA = OC

\(\frac{OA}{2}=\frac{OC}{2}\)

OQ = OS

Similarly, since OB = OD

ie., OR = OP

The diagonals of PQRS bisect each other.

(Since OQ = OS and OR = OP)

Since the diagonal AC is perpendicular to BD, the diagonals of PQRS are also mutually perpendicular bisectors.

therefore PQRS is a rhombus

ii. Area of quadrilateral PQRS = 3 sq. cm.