Question

In a quadrilateral ABCD, $\angle B={90}^{\circ }$ and $\angle D={90}^o$. Then:

• A. $2{AC}^2-{AB}^2={BC}^2+{CD}^2+{DA}^2$
• B. $2{AC}^2={BC}^2+{CD}^2+{DA}^2$
• C. $2{AC}^2-2{AB}^2={BC}^2+{CD}^2+{DA}^2$
• D. $2{AC}^2=2{BC}^2+{CD}^2+{DA}^2+{BC}^2$

Answer 1

The correct option is A $2{AC}^2-{AB}^2={BC}^2+{CD}^2+{DA}^2$

In quadrilateral $ABCD$,

$\angle B=\angle D={90}^{\circ }$
Now, join $AC$
In $△ABC$,
${AC}^2={AB}^2+{BC}^2$
In $△ADC$,
${AC}^2={AD}^2+{CD}^2$
Add both the equations:
$2{AC}^2={AB}^2+{BC}^2+{AD}^2+{CD}^2$

Hence, option A.