Question

AD is median of BC of a triangle ABC and E is mid point of AD .Join AE to F or Join AF where F is a point on AC. Prove that AF=1\3AC.
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Answer 1

Explanation:

Given : A triangle ABC where AD is the median to side BC, E is the mid point of AD. AF is joined such that F is a point on AC.

To Prove : AF = 1/3 (AC)

Construction : Join BE so that BE + EF = BF which meets AC at F. Draw, DG which is parallel to BF.

Proof :

In triangle BFC

• D is the mid point of BC and DG is parallel to BF (By construction)
• Therefore, by converse of Mid Point Theorem, G is the mid point of CF.
• Hence, FG = GC (equation 1)

Now in triangle ADG

• E is the mid point of AD and EF is parallel to DG (By construction, BF is parallel to DG & EF is a part of BF)
• Therefore, by converse of Mid Point Theorem, F is the mid point of AG.
• Hence, AF = FG (equation 2)

From equation 1 and 2

• FG = GC = AF
• But, FG + GC + AF = AC
• it implies, 3(AF) = AC
• it implies, AF = 1/3 (AC)

Hence Proved.

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