Given: ABCD is a quadrilateral in which AD = BC. P, Q, R, S be the mid-points of AB, AC, CD and BD respectively.
To show: PQRS is a rhombus.
Solution:So, we have, a quadrilateral ABCD where AD = BC
And P, Q, R and S are the mid-point of the sides AB, AC, and BD
We need to prove that PQRS is a rhombus.
In ΔBAD, P and S are the mid points of the sides AB and BD respectively,By midpoint theorem which states that the line joining mid-points of a triangle is parallel to third side we get,
PS||AD and PS = 1/2 AD…………(i)
In ΔCAD, Q and R are the mid points of the sides CA and CD respectively,by midpoint theorem we get,
QR||AD and QR = 1/2 AD …………..(ii)
Compare (i) and (ii)
PS||QR and PS = QR
Since one pair of opposite sides is equal and parallel,
Then, we can say that PQRS is a parallelogram…………(iii)
Now, In ΔABC,P and Q are the mid points of the sides AB and AC respectively,by midpoint theorem,
PQ||BC and PQ = 1/2 BC…………..(iv)
And AD = BC …………………………..(v) (given)
Compare equations (i) (iv) and (v), we get,
PS = PQ ………………………………….(vi)
From (iii) and (vi), we get,
PS = QR = PQ Therefore, PQRS is a rhombus.
