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Converse : Triangles with Equal areas and Equal Bases have Equal Corresponding Altitudes.


Given: Area of two triangles ABC and DBC are on the same base BC and are equal. AE and DF are altitudes of triangles ABC and DBC respectively.
To Prove: AE = DF


Proof : ar Δ ABC = ar Δ DBC
× AP × BC = × DQ × BC
Triangles with equal areas and equal bases have equal corresponding altitudes.

Example :

Triangles ABC and DBC are on the same base BC with A, D on the opposite sides of the line BC, such that the ar(ΔABC) = ar(ΔDBC). Show that BC bisects AD. 

Solution :

Given : ΔABC and ΔBCD are on the same base BC. A and D are on opposite sides of BC and ar(ΔABC) = ar(ΔBCD).
To prove: AO = OD.

Construction: Draw perpendiculars AM and DN on BC from A and D respectively.

ar(ΔABC) = ar(ΔBCD) ...(given)
BC x AM = BC x DN
... AM = DN
In the
ΔAMO and the ΔDNO,
AMO = DNO ...(each 90°)
AOM = DON ...(Vertically opposite angle)
AM = DN (Proved above)
...  AO = OD

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