Question

Squares $ABDE$ and $ACFH$ are drawn externally on the sides $AB$ and $AC$, respectively, of a scalene $△ABC.$ Which one of the following is correct$?$

• A. $BH=CE$
• B. $AD=AF$
• C. $BF=CD$
• D. $DF=EH$

Answer 1

The correct option is A $BH=CE$

$△ABC$ is a scalene $△.$
$ACFH$ and $ABDE$ are squares drawn on sides $AC$ and $AB$ respectively.
$\angle BAE=\angle CAH={90}^{\circ }$ (Angle of a square)
$\therefore$ $\angle BAE+\angle BAC=\angle CAH+\angle BAC$
$\Rightarrow$ $\angle CAE=\angle BAH$
In $△$s $EAC$ and $HAB$
$EA=AB$ {sides of same square}
$AC=AH$
$\angle CAE=\angle BAH$ (proved)
$\therefore$ $△EAC\cong △HAB$ (SAS)
$\Rightarrow$ $EC=BH$ (cpct)