Show that the line segments joining the mid points of the opposite sides of a quadrilateral and bisect each other
Answers
Sol : Given ABCD is a quad. P,Q, R and S are the mid points of sides AB BC CD and DA. PR and QS intersect each other at O
To prove : OP=OR, OQ =OS
Proof : In ∆ ABC
P is the mid point of AB
Q is the mid point of BC
Therefore PQ//AC and PQ =1/2AC(by mid point theorem ) --------(1)
In ∆ ADC
S is the mid point of AB and
R is the mid point of DC
Therefore RS//AC and RS=1/2AC ( by mid point theorem) ---------(2)
From eq (1) and (2) we get
PQ//SR and PQ=SR
Therefore PQRS is a parallelogram ( since one pair of opposite side is equal and parallel)
Since the diagonals of a parallelogram bisect each other
Therefore PR and QS bisect each other
Hence OP=OR and OQ=OS
Answer:
Hence proved!
Step-by-step explanation:
From figure:
A Quadrilateral ABCD such that the mid-points of AB,BC,CD,DA are P,Q,R,S.
(i)
In ΔABC,
E and F are the mid-points of AB and BC respectively.
⇒ EF = (1/2) AC and EF || AC {Mid-point theorem}
(ii)
In ΔADC,
H and G are mid-points of AD and CD respectively.
⇒ HG = (1/2) AC and HG || AC
From (i) & (ii), we get
⇒ EF = HG and EF || HG.
(iii)
∴ EFGH is a parallelogram.
The diagonals of a parallelogram bisect each other.
∴ EG and FH bisect each other.
Hence proved!.
Hope it helps!
