Question

In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC.
(vii) If AD = 2 cm, AB = 6 cm, and AC = 9 cm, find AE.
(viii) If AD/BD = 4/5 and EC = 2.5 cm, find AE.
(ix) If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find the value of x.

Answer 1

BASIC PROPORTIONALITY THEOREM (BPT) :  

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

SOLUTION :

7) Given : Δ ABC &  DE || BC , AD = 2 cm, AB = 6 cm, and AC = 9 cm

DB = AB - AD

DB =  6 – 2  

DB = 4 cm

Let AE = x cm

EC = AC - AE

EC = 9 - x

So, AD/DB=AE/ EC

[By using basic proportionality  Theorem]

Then, 2/4= x /(9–x)

4x = 18 – 2x

4x + 2x = 18

6x = 18

x = 18/6  

x = 3 cm

Hence, the value of  x is 3 cm.

8) Given : Δ ABC &  DE || BC ,AD/BD=4/5 and EC = 2.5 cm

So, AD/DB=AE/ EC

[By using basic proportionality  Theorem]

Then, 4/5=AE/2.5

AE = (4×2.5)/5

AE = 4 × .5

AE = 2 cm

Hence, the length of AE is 2 cm.

9) Given : Δ ABC &  DE || BC , AD = x, DB = (x – 2), AE = (x + 2) and EC =( x – 1)

So, AD/DB=AE/ EC

[By using basic proportionality  Theorem]

Then, x/(x–2) = (x+2)/(x–1)

x(x – 1) = (x – 2)(x + 2)

x² – x =  (x)² - (2)²

x² – x =  x² - 4

x² – x² - x =   - 4

-x = - 4

x = 4

Hence, the value of  x is 4 cm.

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Answer 2

Step-by-step explanation:

REFER TO THIS PIC FOR THE SOLUTION ✌️