Question

ABCD is a rectangle. P, Q, R and S are mid-points of sides of the rectangle as shown in the given figure. Prove that PQRS is a rhombus.

[Hint: Join DB. Use mid-point theorem in ΔADB and ΔCDB to show PS = `1/2` DB and PS || to DB, etc.]

Answer 1

Step 1:

S and P are midpoints of AD and AB.

So, PS = `1/2` DB and PS || DB.

Step 2:

Q and R are midpoints of BC and CD.

So, QR = `1/2` DB and QR || DB.

Step 3:

P and Q are midpoints of AB and BC.

So, PQ = `1/2` AC and PQ || AC.

Step 4:

S and R are midpoints of AD and CD.

So, SR = `1/2` AC and SR || AC.

Step 5:

Diagonals of a rectangle are equal AC = DB.

Step 6:

From steps 1, 2, 3, 4 and 5:

`PS = 1/2 DB`

`QR = 1/2 DB`

`PQ = 1/2 AC`

`SR = 1/2 AC`

Since AC = DB, all four segments are equal PS = QR = PQ = SR.