Question

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

Answer 1

Hii mate here is your answer ⤵⤵⤵
first method ❤❤❤
▶Given: ABCD is a quadrilateral in which AD = BC. P, Q, R and S be the midpoints of AB, AC, CD and BD respectively.

▶To prove: PQRS is a rhombus.

▶Proof: By a theorem, 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 Δ ABC, PQ||BC and PQ = BC = DA
In Δ CDA, RQ||DA and RQ = DA
In Δ BDA, SP||DA and SP = DA
In Δ CDB, SR||BC and SR = BC = DA
Therefore SP || RQ, PQ || SR and PQ = RQ = SP = SR.

Hence PQRS is a rhombus.
❤second method ❤❤
Given: AD = BC
To prove: PQRS is a rhombus
Theorem Used:
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.
Prof⤵⤵
In ΔBAD,
P and S are the mid points of sides AB and BD,
So, by midpoint theorem,
PS||AD and PS = 1/2 AD … (i)
In ΔCAD,
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.
hope it helps you ✌✌✌✌✌✌
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