Mathematics
In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.
Mid-point Theorem
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Answer
By mid-point theorem,
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
In △ABD,
Since, P and Q are the mid-points of AB and BD respectively.
PQ || AD
⇒ PQ = × AD …..(1)
In △BCD,
Since, R and Q are the mid-points of DC and BD respectively.
QR || BC
⇒ QR = × BC
⇒ QR = × AD (∵ AD = BC) …..(2)
In △ABC,
Since, P and S are the mid-points of AB and AC respectively.
PS || BC
⇒ PS = × BC
⇒ PS = × AD (∵ AD = BC) …..(3)
In △ADC,
Since, S and R are the mid-points of AC and DC respectively.
SR || AD
⇒ SR = × AD …..(4)
From eq.(1), (2), (3) and (4), we have:
⇒ PQ = SR = QR = PS
Since, PQ || SR (Both are parallel to AD) and QR || PS (both are parallel to BC)
∴ PQRS is rhombus.
Hence, proved that PQRS is a rhombus.
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