(3) In the parallelogram ABCD, the line drawn through a point P on AB,
parallel to BC mots ACat Q. The line through Q.parallel to AB mects
ADAR
Prove that -
RD
Answers
Answer:
Given:- ABCD is ∣∣gm
E is mid point
AP∣∣EC
To prove: BP=2AD
Proof:
In ΔEBC and ΔABP
∠B=∠B (common)
∠PAB=∠CEB (corresponding angle)
∴ By AA congruence
ΔABP≅ΔEBC
So,
EB
AB
=
BC
BP
[properties of similar Δ]
EB
2EB
=
BC
BP
[∵E is midpoint]
1
2
=
AD
BP
[BC=AD opposite side of || gm are equal]
BP=2AD Hence, proved.
To prove: (ii) O is midpoint of AP
proof: In ΔOPC and ΔAPB
∠P=∠P common
∠POC=∠PAB (DC∣∣AB corresponding angles)
By ΔA congruence
ΔOPC≅ΔAPB
So,
AB
OC
=
BP
PC
AB
OC
=
BP
PC
AB
OC
=
2AD
PC
[proved in part (i)]
AB
OC
=
2BC
PC
.......(i) [∵AD=BC]
As BP=2AD
SO, BP=2BC [∵AD=BC]
So,
∵C is the midpoint of BP
Hence, BC=PC
Putting in equation (1)
AB
OC
=
2BC
BC
AB=2OC
DC=2OC (∵AB=DC opposite sides of || gm)
∴O is mid point of CD
Hence, proved.
solution
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Answer:
sorry I didn't know.....