Proofs on Basic Proportionality

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In this section, we shall discuss some more proofs on basic proportionality.

Theorem 1: The line drawn from the mid point of one side of a triangle parallel to another side bisects the third side.

Given : Given ΔABC, D is the mid point of side AB. DE || BC.

Prove that : AE = EC.


statements
Reasons
1) DE || BC 1) Given
2) AD/DB = AE/EC 2) By Basic Proportionality Theorem
3) D is the mid point. 3) Given
4) AD = DB 4) By definition of mid point.
5) AD/DB = 1 5) By division property
6) AE /EC = 1 6) From (2) and (5)
7) AE = EC 7) By cross multiply

Theorem 2 : Prove that the diagonals of a trapezoid(trapezium) divide proportionally.

Given : ABCD is a trapezoid.

Prove that : DE/EB = CE /EA

Construction : Draw EF || BA || CD, meeting AD in F.


Statements
Reasons
1) FE || AB 1) Given (in ΔABD)
2) DE/EB = DF/FA 2) By Basic proportionality theorem
3) FE || DC 3) Given (in ΔCDA)
4) CE/EA = DF/FA 4)By Basic proportionality theorem
5) DE/EB = CE/EA 5) From (2) and (4)

Theorem 3 : Any line parallel to the sides of a trapezium (trapezoid) divides the non-parallel sides proportionally.

Given : ABCD is a trapezoid. DC || AB. EF || AB and EF || DC.

Prove that : AE/ED = BF/FC

Construction : Join AC, meeting EF in G.


Statements
Reasons
1) EG || DC 1) Given (in ΔADC)
2) AE/ED = AG/GC 2) By Basic proportionality theorem
3) GF || AB 3) Given (in ΔABC)
4) AG/GC = BF/FC 4)By Basic proportionality theorem
5) AE/ED = BF/FC 5) From (2) and (4)


Similarity in Triangles

Similarity in Geometry
Properties of similar triangles
Basic Proportionality Theorem(Thales theorem)
Converse of Basic Proportionality Theorem
Interior Angle Bisector Theorem
Exterior Angle Bisector Theorem
Proofs on Basic Proportionality
Criteria of Similarity of Triangles
Geometric Mean of Similar Triangles
Areas of Two Similar Triangles

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