Question

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

Answer 1

SOLUTION :  

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

Proof :  

In ΔABC, P and Q are the mid points of the sides B and AC.

By the midpoint theorem, we get,

PQ || BC & PQ = 1/2 BC………(1)

In ΔADC, Q and R are the mid points of the sides AC and DC.

By the mid point theorem, we get,

QR || AD and QR = 1/2 AD = 1/2 BC

QR = 1/2 BC  …….(2)

[AD = BC]

In ΔBCD,

By the mid point theorem, we get,

RS || BC and RS = 1/2 AD = 1/2 BC

RS = 1/2 BC ………(3)    

[AD = BC]

In ∆BAD, by mid-point theorem

PS || AD and PS = ½ AD= 1/2 BC

PS = 1/2 BC ………….(4)

[AD = BC]

From eqn (1) , (2) , (3) &(4),

PQ = QR = RS= PS

Hence, PQRS is a rhombus.

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