In the figure , ABCD is a parallelogram. P, Q , R , S are the midpoint of the sides of the parallelogram . Then prove that PQ = RS and QR = PS
Answers
Answer:
Step-by-step explanation:
Given : -
ABCD is a parallelogram
P , Q , R & S are the mid points of the sides AB , BC , CD , DA
Required to prove : -
PQ = RS
QR = PS
Theorem used : -
Mid point theorem
This states that ;
The line segments joining the midpoints of the two sides of the triangle is parallel to the third side and also half of the third side .
Solution : -
ABCD is a parallelogram
P , Q , R & S are the mid points of the sides AB , BC , CD , DA
Here,
AC is a diagonal .
Consider ∆ ABC & ∆ ADC
In ∆ ABC & ∆ ADC .
=> AC = AC ( side )
[ Reason : Common side ]
=> AB = CD ( side )
[ Reason : In a parallelogram, opposite sides are equal ]
=> AD = BC ( side )
[ Reason : In a parallelogram , opposite angles are equal ]
By using S.S.S. Congruency rule
∆ ABC ∆ADC
However,
Similarly, if you want to write easily you can write that in a parallelogram a diagonal divides it into 2 Congruency triangles . so ∆ ABC ∆ADC
This implies ;
PQ = ½ AC
RS = ½ AC
[ Reason : Mid point theorem ]
Since, RHS part is equal . Let's equate the LHS part .
Hence,
PQ = RS
Similarly,
Construction : -
Draw the diagonal BD in the ||gm PQRS .
Consider ∆ BCD & ∆ BAD
In ∆ BCD & ∆ BAD
=> BD = BD ( side )
[ Reason : Common side ]
=> BC = AD ( side )
[ Reason : Opposite sides are equal ]
=> CD = AB ( side )
[ Reason : Opposite sides are equal ]
By using S.S.S. Congruency rule ;
∆ BCD ∆BAD
This implies ;
QR = ½ BD
QR = ½ BD PS = ½ BD
[ Reason : Mid point theorem ]
Since,
RHS part is equal . Let's equate the LHS part
Hence,
QR = PS
Hence Proved