8. Prove that in a trapezium, the length of the line segment, joining the mid-points of the non-parallel
sides, is half the sum of the lengths of the parallel sides.
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Step-by-step explanation:
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Step-by-step explanation:
Data: In the trapezium ABCD,
AD || BC, AX = XB and DY = YC
To Prove: (i) XY || AD or XY || BC
(ii) XY = 21(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]
∴ABZA=DCZD [∵ BPT]
∴2AXZA=2DYZD [∵ X & Y are mid points of AB and DC]
∴AXZA=DYZD
⇒XY∣∣AD [∵ Converse of B.P.T.]
(ii) In △ABC,AX=XB [∵ Data]
XP∣∣BC [∵ Proved]
∴AP=PC [∵ Converse of mid point theorem]
∴XP=21BC [∵ Midpoint Theorem]
In △ADC,PY=21AD
By adding, we get XP+PY=21BC+21AD
∴XY=21(BC+AD)
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