ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BO respectively, show that PQRS is a rhombus.
Given, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid points of AB, AC, CD, BD, respectively.
To prove,
PQRS is a rhombus
In ΔABC, P and Q are the mid points of the sides AB and AC respectively
By the midpoint theorem, we get,
PQ∥BC, PQ = 1/2 BC. ---(i)
In ΔADC, Q and R are the mid points of the sides AC and DC respectively
By the mid point theorem, we get,
QR∥AD and QR = 1/2 AD = 1/2 BC (AD = BC) ---(ii)
In ΔBAD,
By the mid point theorem, we get,
PS∥AD and PS = 1/2 AD (AD = BC) ---(iii)
In ΔBCD, R and S are the mid points of the sides CD and BD respectively
By the midpoint theorem, we get,
RS∥BC and SR = 1/2 BC (AD = BC) ---(iv)
From above eqns.
PQ = QR = RS = PS
Thus, PQRS is a rhombus.