Question

Abcd is a quadrilateral in which ad=bc.e,f,g,h are the mid points of ab,bd,cd and ac respectively. Prove that efgh is a rhombus

Answer 1

Thank you for asking this question. Here is your answer:

First of all we will draw a quadrilateral ABCD with AD = BC and join AC, BD, P,Q,R,S are the mid points of AB, AC, CD and BD respectively.

In the triangle ABC, P and Q are mid points of AB and AC respectively.

PQ || BC and PQ = 1/2

BC .. (1)

In ΔADC, QR = 1/2

AD = 1/2

BC ... (2)

Now we will consider ΔBCD,

SR = 1/2

BC.. (3)

In ΔABD,

PS = 1/2

AD = 1/2

BC.. (4)

So from (1), (2), (3) and (4)

we will get

PQ = QR = SR = PS

All sides are equal so PQRS is a Rhombus.

If there is any confusion please leave a comment.