Question

In the given figure, C is midpoint of AB. If $CD=CE$ and $\angle, then which of the following is false?

• A. $ar△APD=ar△EBP$
• B. $ar△ADC=ar△EBC$
• C. $ar△XAC+ar△PYE=ar△DXP+ar△CYB$
• D. none of these

Answer 1

Answer: (C)

$ar△XAC+ar△PYE=ar△DXP+ar△CYB$

Given,

$C$ is the midpoint $⟹AC=BC,CD=CE$

$\angle ACE=\angle BCD$

In $\Delta ACE$ and $\Delta BCD$

$AC=BC$ (common)

$CD=CE$ (given)

$\angle ACE=\angle BCD$ (given)

So by $SAS$ criteria

$\Delta ACE$ is congruent to $\Delta BCD$.

$⟹ar\left(\Delta ACE\right)=ar\left(\Delta BCD\right)$

Subtracting $ar\left(XCYP\right)$ on both side

$\Rightarrow ar\left(\Delta XAC\right)+ar\left(\Delta PYE\right)=ar\left(\Delta DXP\right)+ar\left(\Delta CYB\right)$