Question
For the following statments state whether true (T) or false(F):If O is any point inside a rectangle ABCD then $OA^2 + OC^2 = OB^2 + OD^2$
✓
Answer
Construction: Draw EF || AB through O.
In $\triangle OEA$ and $\triangle OFC$, by pythagoras theorem,
$O A^2=O E^2+A E^2 \text { and } O C^2=O F^2+C F^2$
Adding the two equations, we get
$O A^2+O C^2=O E^2+A E^2+O F^2+C F^2 \ldots \ldots \text { (i) }$
$\triangle OFB$ and $\triangle ODE$, by pythagoras theorem,
$O B^2=O F^2+F B^2 \text { and } O D^2=O E^2+D E^2$
Adding the two equation, we get
$O B^2+O D^2=O F^2+F B^2+O E^2+D E^2$
By Construction since EF || CD,
$D E=C F \text { and } A E=F B$
So, from (i) and (ii), we have
$O A^2+O C^2=O B^2+O D^2$
In $\triangle OEA$ and $\triangle OFC$, by pythagoras theorem,
$O A^2=O E^2+A E^2 \text { and } O C^2=O F^2+C F^2$
Adding the two equations, we get
$O A^2+O C^2=O E^2+A E^2+O F^2+C F^2 \ldots \ldots \text { (i) }$
$\triangle OFB$ and $\triangle ODE$, by pythagoras theorem,
$O B^2=O F^2+F B^2 \text { and } O D^2=O E^2+D E^2$
Adding the two equation, we get
$O B^2+O D^2=O F^2+F B^2+O E^2+D E^2$
By Construction since EF || CD,
$D E=C F \text { and } A E=F B$
So, from (i) and (ii), we have
$O A^2+O C^2=O B^2+O D^2$
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