Question

We get a rhombus by joining the mid-points of the sides of a

Options

• parallelogram

• rhombus

• rectangle

• triangle

Answer 1

We get a rhombus by joining the mid-points of the sides of a rectangle.

It is given a rectangle ABCD in which P,Q,R and S are the mid-points AB,BC,CD and DArespectively.

PQ,QR,RS and SP are joined.

In ΔABC, P and Q are the mid-points AB and BC respectively.

Therefore,

PQ || AC and  `PQ = 1/2 AC` ……(i)

Similarly, In ΔADC,R and S are the mid-points CD and AD respectively.

Therefore,

 SR || AC and `SR = 1/2 AC `……(ii)

From (i) and (ii), we get

PQ || SR and PQ || SR 

Therefore,PQRS is a parallelogram. …… (iii)

Now ABCDis a rectangle.

Therefore,

AD = BC

`1/2 AD = 1/2 BC`

AS = BQ …… (iv)

In ΔAPS and ΔBPQ, we have:

AP = BP (P is the mid point of AB)

∠PAS =∠PBQ  (Each is a right angle)

AS = BQ  (From equation (iv))

So, by SAS congruence criteria, we get:

ΔAPS ≅ ΔBPQ

By Corresponding parts of congruent triangles property we have:

PS = PQ …… (v)

From (iii) and (v) we obtain that PQRS is a parallelogram such that RS = PQ and PQ = QR

Thus, the two adjacent sides are equal.

Thus, PQRS is a rhombus.

Hence the correct choice is (c).