Question

ABCD 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​

Answer 1

Given: ABCD is a rectangle

P, Q, R, S are the mid points of ABCD

To prove: PQRS is a rhombus ( parallelogram and adjacent sides)

Construction: Join AC

Proof:

In triangle ADC,

S is the mid point of AD

R is the mid point of DC

Therefore, SR || AC and SR = 1/2 AC »»» Eq 1

(by midpoint theorem)

In triangle ABC,

P is the mid point of AB

Q is the mid point of BC

Therefore, PQ || AC and PQ = 1/2 AC »»» Eq 2

(by midpoint theorem)

From Eq 1 and Eq 2

SR || PQ, SR=PQ

therefore, PQRS is a parallelogram (a pair of oppo sides equal)

In triangle SAP and in triangle QBP

SA=QB ( oppo sides of a rectangle are equal

1/2 AD =1/2 BC )

<A = <B = 90 degree ( angles of rectangle are 90 degree)

AP = BP ( P is the midpoint )

therefore, triangle SAP =~ triangle QBP ( SAS rule )

PS =PQ ( cpct )

therefore, PQRS is a rhombus [ a parallelogram with a pair of adjacent sides equal]

Hope it heps you !!!!

Answer 2

ur answer is attached....