In the adjoining figure, DE || BC. Prove that: 1. ar( )=ar( ) 2. … - Vidyadip

Question

In the adjoining figure, DE || BC. Prove that:

$\text{ar}(\triangle\text{ACD})=\text{ar}(\triangle\text{ABE})$
$\text{ar}(\triangle\text{OCE})=\text{ar}(\triangle\text{OBD}).$

✓

Answer

$\triangle\text{DEC}$ and $\triangle\text{DEB}$ lies on the same base and between the same parallel lines.So, $\text{ar}(\triangle\text{DEC})=\text{ar}(\triangle\text{DEB})\dots(1)$

On adding $\text{ar}(\triangle\text{ADE})$ in both sides of equation (1), we get:

$\text{ar}(\triangle\text{DEC})+\text{ar}(\triangle\text{ADE})=\text{ar}(\triangle\text{DEB})+\text{ar}(\triangle\text{ADE})$
$\Rightarrow\ \text{ar}(\triangle\text{ACD})=\text{ar}(\triangle\text{ABE})$

On adding $\text{ar}(\triangle\text{ODE})$ in both sides of equation (1), we get:

$\text{ar}(\triangle\text{DEC})-\text{ar}(\triangle\text{ODE})=\text{ar}(\triangle\text{DEB})-\text{ar}(\triangle\text{ODE})$
$\Rightarrow\ \text{ar}(\triangle\text{OCE})=\text{ar}(\triangle\text{OBD}).$

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