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High School Mathematics - 2
Basic Proportionality Theorem or Thale's Theorem

A straight line drawn parallel to a side of a triangle, divides the other two sides proportionally.
Data: Triangle ABC, D is a point on AB and the line DE is parallel to BC
To prove: AD/BD = BE/CE

Consider triangle ABC and triangle BDE
StepStatement Reason
1ang BAC = ang BDECorresponding angles because DE ll AC
2ang BCA = ang BEDCorresponding angles because DE ll AC
3ang DBE = ang ABCCommon angle
4triangle ABC lll triangle BDEAAA postulate on similarity
5AB/BD = BC/BEcorresponding sides are proportional
6(BD+DA)/BD =(BE+EC)/BESplit AB and AC
71+AD/BD = 1+CE/BESimplification
8AD/BD = CE/BESubtract 1 from either sides
10AD/BD= BE/CEReciprocating

Converse
We shall try to prove the converse of basic proportionality theorem.
Let us consider triangle ABC shown in the figure below.

Divide the sides of the triangle AB and AC into any number of equal parts.
By measuring the angles B1 and B we observe that they are equal.
But B1 and B are corresponding angles.
Hence B1C1 ll BC.
Similarly
AB2/B2 B= AC2/C2C = 2/3 and B22C2 ll BC
AB3/B3B =AC3/C3C= 3/2 and B3C3 ll BC
AB4/B4B = AC4/C4C and B4C4 ll BC

Example: In the figure, find the length of PS given that ST ll QR.

Let PS = x cm
As ST ll QR
PS/QS = RT/PT
x/3 = 3/2
x = 9/2
Answer: PS = 4.5 cms


Directions: Answer the following questions.

Name: ___________________

Date:___________________

High School Mathematics - 2
Basic Proportionality Theorem or Thale's Theorem