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.
Given: AD = BC
To prove: PQRS is a rhombus
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
P and S are the mid points of sides AB and BD,
So, by midpoint theorem,
PS||AD and PS = 1/2 AD … (i)
R and Q are the mid points of CD and AC,
So, by midpoint theorem
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 as well as parallel then
PQRS is a parallelogram … (iii)
Now, In ΔABC, by midpoint theorem
PQ||BC and PQ = 1/2 BC … (iv)
And Ad = BC … (v)
Compare equations (i) (iv) and (v)
PS = PQ … (vi)
From (iii) and (vi)
Since PQRS is a parallelogram with PS = PQ then PQRS is a rhombus.
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