**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.

**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 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:**

**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 TX^{2} = TB × TC.

**Solution: **In ΔTXM, we have

**Example 3: **In fig., EF || AB || DC. Prove that (frac{AE}{ED}=frac{BF}{FC}).

**Solution: **We have, EF || AB || DC

**Example 4: **In figure, ∠A = ∠B and DE || BC. Prove that AD = BE

**Solution:**

**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:**

**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.

**Example 7: **In fig. DE || BC and CD || EF. Prove that AD^{2} = AB × AF.

**Solution:**

**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:**

**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:**

**Example 10: **In a given ∆ABC, DE || BC and (frac{AD}{DB}=frac{3}{4}). If AC = 14 cm, find AE.

**Solution:**

**Example 11: **In figure, DE || BC. Find AE.

**Solution:**

**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

**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:**

**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:**