Question -:
Prove that the line segment joining midpoints of non - parallel sides of a trapezium is parallel to parallel sides.
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Answer:
In the trapezium ABCD,
AD || BC, AX = XB and DY = YC
To Prove: (i) XY || AD or XY || BC
(ii) XY =
2
1
(AD + BC)
Construction: Extend BA and CD to meet at Z.
Join A and C. Let it cut XY at P
Proof: (i) In △ZBC,AD∣∣BC [∵ Data]
∴
AB
ZA
=
DC
ZD
[∵ BPT]
∴
2AX
ZA
=
2DY
ZD
[∵ X & Y are mid points of AB and DC]
∴
AX
ZA
=
DY
ZD
⇒XY∣∣AD [∵ Converse of B.P.T.]
(ii) In △ABC,AX=XB [∵ Data]
XP∣∣BC [∵ Proved]
∴AP=PC [∵ Converse of mid point theorem]
∴XP=
2
1
BC [∵ Midpoint Theorem]
In △ADC,PY=
2
1
AD
By adding, we get XP+PY=
2
1
BC+
2
1
AD
∴XY=
2
1
(BC+AD)
Step-by-step explanation:
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