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.
| 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)
⇒ 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