Question

prove that the line segment joining the mid points of two sides of a triangle is parallel to the third side and it is half of its third side​

Answer 1

Step-by-step explanation:

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Answer 2

Proof: Through C, draw a line parallel to BA, and extend DE such that it meets this parallel at F, as shown below:

Midpoint Theorem

Compare  

Δ A E D  with  Δ C E F :  

1. AE = EC (E is the midpoint of AC)  

2.  ∠ DA E  =  ∠ F CE  (alternate interior angles)

3.  ∠ D E A  =  ∠ F E C  (vertically opposite angles)

By the ASA criterion, the two triangles are congruent. Thus, DE = EF and AD = CF. But AD is also equal to BD, which means that BD = CF (also, BD || CF by our construction). This implies that BCFD is a parallelogram. Thus,

1. DF || BC è DE || BC

2. DE = EF = ½(DF) = ½(BC) èDE = ½(BC)