Basic Proportionality Theorem

Basic Proportionality Theorem (Thales theorem): If a line is drawn parallel to one side of a triangle intersecting other two sides, then it divides the two sides in the same ratio.

In ∆ABC , if DE || BC and intersects AB in D and AC in E then     AD       AE    ---- = ------     DB      EC

Proof on Thales theorem :

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

Given : In ∆ABC , DE || BC and intersects AB in D and AC in E.

Prove that : AD / DB = AE / EC

Construction : Join BC,CD and draw EF ┴ BA and DG ┴ CA.

Statements	Reasons
1) EF ┴ BA	1) Construction
2) EF is the height of ∆ADE and ∆DBE	2) Definition of perpendicular
3)Area(∆ADE) = (AD .EF)/2	3)Area = (Base .height)/2
4)Area(∆DBE) =(DB.EF)/2	4) Area = (Base .height)/2
5)(Area(∆ADE))/(Area(∆DBE)) = AD/DB	5) Divide (4) by (5)
6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC	6) Same as above
7) ∆DBE ~∆DEC	7) Both the ∆s are on the same base and between the same || lines.
8) Area(∆DBE)=area(∆DEC)	8) If the two triangles are similar their areas are equal
9) AD/DB =AE/EC	9) From (5) and (6) and (7)

Some solved examples :
1) In the given figure, PQ || MN. If KP / PM = 4 /13 and KN = 20.4 cm.
Find KQ.

Solution :
In Δ KMN,

PQ || MN

∴ KP / PM = KQ / QN ( By BPT theorem)

⇒ KP / PM = KQ / ( KN – KQ)

⇒ 4 / 13 = KQ / ( 20.4 – KQ)

⇒ 4( 20.4 - KQ ) = 13 KQ ( cross multiply )

⇒ 81.6 – 4KQ = 13 KQ

⇒ 17KQ = 81.6

⇒ KQ = 81.6 / 17

∴ KQ = 4.8 cm.

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2) In the figure given below, DE ||BC. If AD = x cm, DB = x-2 cm,
AE = x-1 cm, then find the value of x.



Solution :
In triangle ABC ,

DE || BC

AD /DB = AE /EC ( by basic proportionality theorem)

⇒ x /(x - 2) = (x + 2) /(x - 1)

⇒ x (x - 1) = (x - 2)(x + 2) (by cross multiplication)

⇒ x² - x = x² - 4

⇒ -x = -4

∴ x = 4

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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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