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# In Fig. 9 24, 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) equals ar (ABD).

Q : 4    In Fig. $\small 9.24$,  ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that  $\small ar(ABC)=ar(ABD)$.

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We have, $\Delta$ABC and $\Delta$ABD on the same base AB. CD is bisected by AB at point O.
$\therefore$ OC = OD
Now, in $\Delta$ACD, AO is median
$\therefore$ ar ($\Delta$AOC) = ar ($\Delta$AOD)..........(i)

Similarly, in $\Delta$BCD, BO is the median
$\therefore$ ar ($\Delta$BOC) = ar ($\Delta$BOD)............(ii)

Adding equation (i) and eq (ii), we get

ar ($\Delta$AOC) + ar ($\Delta$BOC) =  ar ($\Delta$AOD) + ar ($\Delta$BOD)

$\small ar(ABC)=ar(ABD)$

Hence proved.

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