Q9 of 70 Page 139

In Δ ABC, such that . Prove that X, Y, Z are the mid-points of , , respectively.


Given: In ∆ABC, X∊BC, Y∊CA, Z∊AB.


Also, XY||AB, YZ||BC and ZX||AC.


To prove: X,Y and Z are the mid-points of BC, CA and AB respectively.


Proof: In ∆ABC, YZ||BC.


…(1)


In ∆ABC, XY||AB.


…(2)


From (1) and (2),


…(3)


In ∆ABC, ZX||AC.


…(4)


From (3) and (4),



AZ2 = BZ2


AZ = BZ


Hence, A-Z-B and AZ = BZ.


Thus, Z is the mid-point of AB.


Similarly, Y is the mid-point of AC and X is the midpoint of BC.


Therefore, X, Y and Z are the mid-points of BC, CA and AB respectively.


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