Question

If D and E are respectively,the points on the sides AB and AC of a triangle ABC,such that AD = 6cm,BD= 9cm,AE= 8cm and EC = 12cm,then show that DE || BC

Answer 1

Given,  

AE =  6 CM

EC = 9 CM  

AD = 8 CM  

DB = 12 CM  

Clearly,

AD/DB = 8/12 =  2/3

and,

AE/ EC = 6/9 = 2/3

Hence,

AE/EC = AD/BD

We know that, if  a line   intersects  two sides  of triangle in same  ratio of sides, Then the intersecting  line  will be half  of  third  side  of  triangle and parallel   to third  side.  

Hence,  

DE || BD  

And,

DE  = 1/2 * BC

BC = 2DE

Answer 2

As triangle ADE  ~ ABC

AD/AB = DE/BC

8/8+12 = DE/BC

8/20 = DE/BC

2/5 = DE/BC

BC = 5/2 DE

AD = 6 cm; BD = 9 cm

AE = 8 cm  and CE = 12 cm

Now we have:

AD/BD = 6/9 = 2/3

AE/CE = 8/12 = 2/3

So AD/BD = AE/CE

We know that if a line intersects any two sides of a triangle in equal ratio then the line is parallel to the third side.

DE = BC