prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of lengths of the parallel sides
Answers
Given : A trapezium and a line join mid points of non parallel sides
To find : prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of lengths of the parallel sides
Solution:
Let say ABCD is trapezium with AB ║ CD
M & N are mid point ( median ) of BC & AD
Now lets draw MP ║ AB ( P is point on AC)
now in Δ ABC
as MP ║ AB
=> CP/AP = CM/BM
now CM = BM ( as M is mid point)
=> CP = AP
also in Δ ABC
MP/AB = CM/BC
=> MP/AB = 1/2
=> MP = AB/2
Now in Δ ACD
AN/ND = AP/CP = 1
=> NP ║ CD
MP ║ AB
=> P lies of MN
=> MN ║ AB ║ CD
QED
Hence proved
median of a trapezium is parallel to the parallel sides of the trapezium
also in Δ ACD
PN /CD = AN/AD
=> PN/CD = 1/2
=> PN = CD/2
MN = MP + PN = AB/2 + CD/2
=> MN = (1/2)(AB + CD)
QED
Hence proved
length is half of the sum of lengths of the parallel sides
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