Question

In the given figure ABCD is a rectangle P, Q, R and S are the mid points of the
sides AB, BC, CD and AD respectively.
(0) Prove that PQRS is a rhombus.
(ii) If the length of AC is 6 cm, find the perimeter of PQRS.​

Answer 1

Answer:

I) PQRS is Rhombus

II) perimeter of PQRS = 12cm

Step-by-step explanation:

Please see the solution.

Answer 2

Proved that PQRS is a rhombus.

Perimeter of PQRS is 12 cm.

(i) To prove : PQRS is a rhombus

(ii) To find : Perimeter of PQRS

Given :

(i) PQRS is a rhombus :

From the figure,

ABCD is a rectangle.

P, Q, R, S are the mid point of AB, BC, CD and AD respectively.

By applying mid point theorem,

SR = PQ = $\frac{1}{2}$ AC   -----> ( 1 )

Now, join BD

By applying mid point theorem,

QR = PS = $\frac{1}{2}$ BD   -----> ( 2 )

By using Pythagoras theorem,

SR² = DS² + DR²      

SR = $\sqrt{DS^2+DR^2}$  = $\sqrt{CQ^2+CR^2}$ = $\sqrt{QR^2}$ = QR  -----> ( 3 )

From ( 1 ), ( 2 ) and  ( 3 )

PQ = QR = RS = SP  

Here, opposite sides are parallel.

Hence, PQRS is a rhombus.

(ii) Perimeter of PQRS :

Length of AC = 6 cm.    

From equation ( 1 ),

SR = PQ = $\frac{1}{2}$ AC              

              = $\frac{1}{2} \times6 = 3 cm$

Hence, a = 3 cm

Perimeter of rhombus  = 4 a = 4 × 3 = 12 cm.

Hence, perimeter of rhombus is 12 cm.

To learn more...

1. ABCD is a rectangle and P, Q, R and S are mid points of the sides AB, BC, CD and DA respectively. show that the quadrilateral PQRS is a rhombus

brainly.in/question/5807421    

2. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle

brainly.in/question/1471371