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In figure below, \( \mathrm{ABC} \) and \( \mathrm{ABD} \) are two triangles on the same base \( \mathrm{AB} \). If line- segment \( \mathrm{CD} \) is bisected by \( \mathrm{AB} \) at \( \mathrm{O} \), show that \( \operatorname{ar}(\mathrm{ABC})=\operatorname{ar}(\mathrm{ABD}) \).
Given:
$ABC$ and $ABD$ are two triangles on the base $AB$.
Line segment $CD$ is bisected by $AB$ at $O$.
To do:
We have to show that $ar(\triangle ABC) = ar(\triangle ABC)$.
Solution:
$CO = OD$
Draw $CL \perp AB$ and $DM \perp AB$
In $\Delta CLO$ and $\triangle D M O$,
$\angle \mathrm{L}=\angle \mathrm{M}$
$\mathrm{CO}=\mathrm{OD}$
$\angle \mathrm{COL}=\angle \mathrm{DOM}$ (Vertically opposite angles)
Therefore, by AAS axiom,
$\Delta CLO \cong \triangle D M O$
This implies,
$\mathrm{CL}=\mathrm{DM}$ (CPCT)
$ar(\triangle \mathrm{ABC})=\frac{1}{2}\times$ Base $\times$ Height
$=\frac{1}{2} \times \mathrm{AB} \times \mathrm{CL}$.........(i)
$ar(\triangle \mathrm{ADB})=\frac{1}{2} \times \mathrm{AB} \times \mathrm{DM}$
$=\frac{1}{2} \mathrm{AB} \times \mathrm{CL}$...........(ii) (Since $\mathrm{DM}=\mathrm{CL}$)
(Proved)
From (i) and (ii), we get,
$ar(\triangle \mathrm{ABC})=ar(\triangle \mathrm{ABD})$
Hence proved.