In the figure LMNO, is a trapezium in which LM is parallel to side ON and P of the midpoint of side LO. If Q us a point on the side MN such that segment PQ ya parallel to side ON prove that Q is the mud point of MN and PQ=1/2(LM+ON).
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Answers
Given : trapezium LMNO in which LM is parallel to side ON and
P is the midpoint of side LO .
Q is any point on side MN such that segment PQ is parallel to side ON .
To Find : Prove that Q is the midpoint of MN and PQ =1/2(LM + ON).
Solution:
PQ || ON
LM || ON
Hence PR || ON || LM as R is on PQ
if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
line joining the mid-point of two sides of a triangle is equal to half the length of the third side
in ΔLON PR || ON and P is mid point of LO
Hence R is mid point of LN
as LP/OP = LR/NR = 1
and PR = (1/2) ON
in ΔLMN PR || LM
R is mid point of LN Hence Q is mid point of MN
as NR/LR = NQ/QM = 1
so QR = (1/2) LM
PR + QR = (1/2) ON + (1/2) LM
=> PQ = (1/2) ( LM + ON)
QED
Hence Proved
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