Basic Proportionality Theorem

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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.

-----------------------------------------------------------------
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)

⇒ x
2 - x = x 2 - 4

⇒ -x = -4

∴ x = 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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