ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that ar(ABC) = ar (ABD). In Fig. 9.24.
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Consider the triangle ACD:
Line-segment CD is bisected by AB at O.
AO is the median of ΔACD.
Therefore, Area (ACO) = Area (ADO) ———Equation 1.
Considering triangle BCD:
BO is the median.
Therefore, Area (BCO) = Area (BDO)———Equation 2.
Adding equation 1 and 2, we get:
=>Area (ACO) + Area (BCO) = Area (ADO) + Area (BDO).
=> Area(ABC) = Area(ABD).
Therefore, Area (ABC) = Area (ABD).
Line-segment CD is bisected by AB at O.
AO is the median of ΔACD.
Therefore, Area (ACO) = Area (ADO) ———Equation 1.
Considering triangle BCD:
BO is the median.
Therefore, Area (BCO) = Area (BDO)———Equation 2.
Adding equation 1 and 2, we get:
=>Area (ACO) + Area (BCO) = Area (ADO) + Area (BDO).
=> Area(ABC) = Area(ABD).
Therefore, Area (ABC) = Area (ABD).
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