Basic Proportionality Theorem or Thales Theorem

Basic Proportionality Theorem or Thales Theorem

Statement: If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
Given: A triangle ABC in which DE || BC, and intersects AB in D and AC in E.
Basic-Proportionality-Theorem
Basic-Proportionality-Theorem-1
Basic-Proportionality-Theorem-2
Basic-Proportionality-Theorem-3

Converse of Basic Proportionality Theorem

Statement: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
Given: A DABC and a line l intersecting AB in D and AC in E,
Basic-Proportionality-Theorem-4
Basic-Proportionality-Theorem-5

Basic Proportionality Theorem Example Problems With Solutions

Example 1:    D and E are points on the sides AB and AC respectively of a ∆ABC such that DE || BC.
Find the value of x, when
(i) AD = 4 cm, DB = (x – 4) cm, AE = 8 cm and EC = (3x – 19) cm
(ii) AD = (7x – 4) cm, AE = (5x – 2) cm,
DB = (3x + 4) cm and EC = 3x cm.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-1

Example 2:    Let X be any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA meeting CA, BA in M, N respectively; MN meets BC produced in T, prove that TX2 = TB × TC.
Solution:    In ΔTXM, we have
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-2

Example 3:    In fig., EF || AB || DC. Prove that (frac{AE}{ED}=frac{BF}{FC}).
Solution:    We have, EF || AB || DC
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-3
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-3-1

Example 4:    In figure, ∠A = ∠B and DE || BC. Prove that AD = BE
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-4
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-4-1

Example 5:    In fig., DE || BC. If AD = 4x – 3, DB = 3x – 1, AE = 8x – 7 and EC = 5x – 3, find the value of x.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-5

Example 6:    Prove that the line segment joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.
Solution:    
Given : A quadrilateral ABCD in which P, Q, R, S are the midpoints of AB, BC, CD and DA respectively.
To prove: PQRS is a parallelogram.
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-6

Example 7:    In fig. DE || BC and CD || EF. Prove that AD2 = AB × AF.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-7

Example 8:    Ex.8 In the given figure PA, QB and RC each is perpendicular to AC such that PA = x,
RC = y, QB = z, AB = a and BC = b. Prove that (frac{1}{x}+frac{1}{y}=frac{1}{z}).
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-8
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-8-1

Example 9:    In fig., LM || AB. If AL = x – 3, AC = 2x, BM = x – 2 and BC = 2x + 3, find the value of x.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-9

Example 10:    In a given ∆ABC, DE || BC and (frac{AD}{DB}=frac{3}{4}). If AC = 14 cm, find AE.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-10

Example 11:    In figure, DE || BC. Find AE.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-11

Example 12:    In figure, ABC is a triangle in which AB = AC. Points D and E are points on the sides AB and AC respectively such that AD = AE. Show that the points B, C, E and D are concyclic.
Solution:    In order to prove that the points B, C, E and D are concyclic, it is sufficient to show that
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-12
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-12-1

Example 13:    In fig., (frac{AD}{DB}=frac{1}{3}text{   and   }frac{AE}{AC}=frac{1}{4}). Using converse of basic proportionality theorem, prove that DE || BC.
Solution:
Basic-Proportionality-Theorem-or-Thales-Theorem-Example-13

Example 14:    Using basic proportionality theorem, prove that the lines drawn through the points of trisection of one side of a triangle parallel to another side trisect the third side.
Solution: