Question

i. prove PQRS is a rhombus
ii. if the length of AC is 6 cm find the perimeter of PQRS​

Answer 1

i. prove PQRS is a rhombus

Consider the given figure while going through the following steps.

Given,

SR ║ PQ  ...........(1)

SP ║ RQ .........(2)

Using mid-point theorem, we get,

SP = PQ = 1/2 AC

QR = PS = 1/2 BD

Now consider,

In Δ SDR and Δ QCR,

SR² = SD² + DR²

= QC² + CR²

= QR²

⇒ SR = QR ...........(3)

Now consider,

In Δ PAS and Δ QBP,

PS² = AS² + AP²

= QB² +BP²

= QP²

⇒ PS = QP ..........(4)

Using (1), (2), (3) and (4), we get,

Opposite sides are parallel and equal in length.

Therefore, it is proved that PQRS is a rhombus.

ii. if the length of AC is 6 cm find the perimeter of PQRS​

Given,

AC = 6 cm

from (i), we have,

SP = PQ = 1/2 AC

SP = PQ = 1/2 × 6 = 3

SP = PQ = QR = RS = 3 cm

The perimeter of rhombus PQRS = 4 × side

= 4 × 3

= 12 cm

Hence the perimeter of PQRS.