Basic Proportionality Theorem ( Thales Theorem ) and its converse
Theorem: If a line is drawn parallel to one side of a triangle to intersectthe other two sides in distinct points, the other two sides are divided in the same ratio.
Given: A triangle ABC and DE || BC
Construction: Join BE and CD and draw DM AC and EN AB
Proof: .......(1) ......(2) .....(3) .....(4) Using Eq 1, 2, 3 and 4 ........(5) ........(6) Now BDE and DEC are on the same base DE and between the same parallel BC || DE So ar(BDE) = ar( DEC) Using the relation in Eq 5 and 6 we get Hence Proved |
Converse of the basic proportionality theorem is also true,
Theorem If a line divides any twosides of a triangle in the same ratio, the line is parallel to the third side. Given : a triangle ABC and That is, if in fig. , To Prove : Construction: Draw DF || BC Proof: If DF || BC then ...... (1) But we are given ...............(2) Then from 1 and 2 we get This is possible only when F and E coincide. This implies DE || BC |
Example :.
Solution , [ Basic Proportionality Theorem]
As |
In the following figure, if PQ || CB and PE || CD, then which of the following is true? | |||
Right Option : D | |||
View Explanation |
, D and E are points on side AB and AC respectively such that DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm then AC = ______________ | |||
Right Option : B | |||
View Explanation |
Find AB, when in so that AD = 2.4 cm, AE = 3.2 cm and EC = 4.8 cm. | |||
Right Option : A | |||
View Explanation |
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