Question

For each of the following statements state whether true(T) or false (F) 

The length of the line segment joining the midpoints of any two sides of a triangles is equal to half the length of the third side.

Answer 1

True
Suppose ABC is a triangle and M, N are 

 

Construction: DE is expanded to F such that EF = DE
To proof = DE =` 1/2`𝐵𝐶
Proof: In ΔADE and ΔCEF
AE = EC (E is the mid point of AC)
DE = EF (By construction)
AED = CEF (Vertically Opposite angle)
By SAS criterion, ΔADE ~= ΔCEF
CF = AD (CPCT)
⟹ BD = CF
∠𝐴𝐷𝐸= ∠𝐸𝐹𝐶 (𝐶𝑃𝐶𝑇)
Since, ∠𝐴𝐷𝐸 𝑎𝑛𝑑 ∠𝐸𝐹𝐶 𝑎𝑟𝑒 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝑎𝑛𝑔𝑙𝑒
Hence, AD ‖ CF and BD ‖ CF
When two sides of a quadrilateral are parallel, then it is a parallelogram
∴ DF = BC and BD ‖ CF
∴BDFC is a parallelogram
Hence, DF = BC
⟹ DE + EF = BC 

⟹ 𝐷𝐸=`1/2` 𝐵𝐶