Question

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.  

 

Answer 1

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

So, PQ || BC, and PQ = `1/2 `BC               ..................(1) 

Similarly, in ΔADC,                                   .................(2) 

Now, in ΔBCD, SR = `1/2` BC                 .........................(3) 

Similarly, in ΔABD, PS = `1/2` AD=`1/2` BC         ...............(4) 

Using (1), (2), (3), and (4).
PQ = QR = SR = PS
Since, all sides are equal
Hence, PQRS is a rhombus.