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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