Question

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

(a) square

(b) rectangle

(c) trapezium

(d) none of these

Answer 1

Figure is given as :

A rhombus ABCD is given in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively.

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

Therefore,

and ……(i)

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

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram. …… (iii)

Now, we shall find one of the angles of a parallelogram.

Since ABCD is a rhombus

Therefore,

(Sides of rhombus are equal)

(P and Q are the mid-points AB and BC respectively)

In , we have

(Angle opposite to equal sides are equal)

Therefore, ABCD is a rhombus

…… (iii)

Also,

…… (iv)

Now, in and , we have

[From (iii)]

[From (iv)]

And ( is a parallelogram)

So by SSS criteria of congruence, we have

By Corresponding parts of congruent triangles property we have:

…… (v)

Now,

And

Therefore,

From (ii), we get

From (v), we get

Therefore, …… (vi)

Now, transversal PQ cuts parallel lines SP and RQ at P and Q respectively.

[Using (vi)]

Thus, is a parallelogram such that .

Therefore, is a rectangle.

Hence the correct choice is (b).