If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in area.

Question:

If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in area. If G is the mid-point of the median AD, prove that ar(ΔBGC) = 2ar(ΔAGC).

Solution:

Draw AM ⊥ BC

Since, AD is the median of ΔABC

∴ BD = DC

⇒ BD = AM = DC × AM

⇒ (1/2)(BD × AM) = (1/2)(DC × AM)

⇒ ar(ΔABD) = ar(ΔACD) ⋅⋅⋅⋅⋅⋅ (1)

In ΔBGC, GD is the median

⇒ ar(ΔBGD) = ar(ΔCGD) ⋅⋅⋅⋅⋅⋅ (2)

In ΔACD, CG is the median

⇒ ar(ΔAGC) = ar(ΔCGD) ⋅⋅⋅⋅⋅ (3)

From (2) and (3) we have,

ar(ΔBGD) = ar(ΔAGC)

But, ar(ΔBGC) = 2ar(ΔBGD)

⇒ ar(ΔBGC) = 2ar(ΔAGC)

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