## EXAMPLE PROBLEMS ON BASIC PROPORTIONALITY THEOREM

Basic Proportionality Theorem :

If a straight 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.

## Example Problems on Basic Proportionality Theorem

Example 1 :

In a triangle ABC,D and E are points on the sides AB and AC respectively such that DE is parallel to BC.

(i)  If AD = 6 cm, DB = 9 cm and AE = 8 cm, then find AC.

Solution :

In the given triangle ABC the sides DE is parallel to the side BC. By using “Thales theorem”, we get

(6/9)  =  (8/EC)

EC  =  (9  8)/6

EC  =  (3  4)

EC  =  12 cm

Example 2 :

In triangle ABC, if AD = 8 cm, AB = 12 cm and AE = 12 cm, then find CE.

Solution :

In the given triangle ABC the sides DE is parallel to the side BC. By using “Thales theorem”, we get

12  =  8 + DB

DB  =  4 cm

(8/4)  =  (12/EC)

EC  =  (12  4)/8

EC  =  48/8

EC  =  6 cm

Example 3 :

In triangle ABC, if AD = 4 x – 3, BD = 3 x – 1, AE = 8 x – 7 and EC = 5 x – 3, then find the value of x.

Solution :

In the given triangle ABC the sides DE is parallel to the side BC. By using “Thales theorem”, we get

(4 x – 3)/(3 x – 1)  =  (8 x – 7)/(5 x – 3)

(4 x – 3) (5 x – 3)  =  (8 x – 7) (3 x – 1)

20 x ² – 12 x – 15 x + 9  =  24 x ² – 8 x – 21 x + 7

20 x ² – 24 x + 9  =  8 x ² – 29 x + 7

20 x ² - 24 x ² – 27 x + 29 x + 9 – 7  =  0

- 4 x ² + 2 x + 2  =  0

Multiply by - 2

2 x ² – x - 1  =  0

(x – 1) (2 x + 1)  =  0

 x - 1  =  0x  =  1 2x + 1  =  02x  =  -1x  =  -1/2

So the value of x is 1.

After having gone through the stuff given above, we hope that the students would have understood, examples on basic proportionality theorem.

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