Question

In the following figure, the diameter of circle is $3cm$, $AB$ and $MN$ are two diameters such the $MN$ is perpendicular to $AB$. In addition, $CG$ is perpendicular to $AB$ such that $AE:EB=1:2$ and $DF$ is perpendicular to $MN$ such that $NL:LM=1:2$. The length of $DH$ in $cm$ is:-

• A. $2\sqrt{2}-1$
• B. $\frac{2\sqrt{2}-1}{2}$
• C. $\frac{3\sqrt{2}-1}{2}$
• D. $\frac{2\sqrt{2}-1}{3}$

Answer 1

The correct option is B $\frac{2\sqrt{2}-1}{2}$

$AB=MN=3$cm

$\frac{AE}{EB}=\frac{NL}{LM}=\frac{1}{2}$

$AB=3$cm

$\Rightarrow AE+EB$

$\Rightarrow 3$

$AE+2AE=3$

$AE=1$cm

$NL=1$cm

$OA=\frac{AB}{2}=1.5$cm

$ON=\frac{MN}{2}=1.5$cm

$OE=OA-AE=1.5-1=0.5$cm

$OL=ON-LN=0.5$cm

Quatrilateral EOLG is square

$HL=EO=0.5$cm

Join OD

$\Delta OLD,$

$O{D}^2=O{L}^2+D{L}^2$

$D{L}^2=(1.5{)}^2-(0.5{)}^2$

$DL=\sqrt{2}cm$

$DH=DL-HL-\sqrt{2}-\frac{1}{2}=\frac{2\sqrt{2}-1}{2}$.