Question

Prove that a line joining the midpoints of any two sides of a triangle is parallel to the third side. (Using converse of basic proportionality theorem)

Answer 1

 Given that ABC is a traingle and D and E are mid points of traingle. Construction : Join C and D , B and E. Prove that : DE // BC.Proof :  In ΔADE = ΔBDE AD = BD ( D is mid point) DE = DE ( same height) Area of ΔADE = Area of ΔBDE -------------(1) In ΔADE = ΔCDE AE = EC ( E is mid point) DE = DE ( same height) Area of ΔADE = Area of ΔCDE -------------(2)from (1) and (2) we get  Area of ΔBDE = Area of ΔCDE ∴ DE // BC.

Answer 2

Given:

ΔABC

Mid points = D and E of sides AB and AC

AD=BD and AE=EC.

To Find:

DE || BC

Solution:

In ΔABC ,

Since, D is the mid point of AB

Therefore, AD=DB

= AD/BD = 1 --- eq 1

Similarly, E is the mid-point of AC (Given)

Therefore, AE=EC

= AE/EC --- eq 2

From equation 1 and 2 we will get -

AD/BD = AE/EC

Therefore, DE || BC ( By CBPT)

Answer: Since, DE || BC line joining mid points of any two sides of a triangle is parallel to the third side