Proofs on Basic Proportionality

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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